Riddell and Taylor asked if, amongst the partitions of n into distinct primes, the one with maximum product was necessarily one of those with the greatest number of components. The answer is no. Selfridge found this example: 319 = 2+3+5+7+11+13+17+23+29+31+37+41+47+53. 319 = 3+5+11+13+17+19+23+29+31+37+41+43+47. Where the product of the 13 terms in the second sum is greater than that of the 14 terms in the first sum.
It is obviously natural to wonder if there are other solutions. Sequence A053020 in the Encyclopedia of Integer Sequences is thus defined to be the sequence of solutions, in increasing order.
Silvio Sergio and I, working independently, found that 319 is the smallest solution.
Here's some sample output from my program. Please let me know if you find any mistakes.
Here is my Perl program for generating solutions to this question:
sub solution {
my $n=shift;
if ($n<2) {
return (($n==1)?$zero:$one,0,0);
}
my $index=shift;
my $firstnum=$primes[$index];
if (defined $early[$n] && $firstnum<$early[$n]) {
return ($zero,0,0);
}
if ($sumprimes{$firstnum}<$n) {
return ($zero,0,0);
}
if (defined $late[$n]) {
if ($firstnum>$late[$n]) {
return @{$a{"$n $late[$n]"}};
} else {
while (!exists $a{"$n $firstnum"}) {
$firstnum=$primes[++$index]
}
return @{$a{"$n $firstnum"}}
}
}
#if ($n<$magic) {
my $string="$n $firstnum";
if (exists $a{$string}) {
return @{$a{$string}};
}
# }
my ($a,$b,$c) = solution ($n,$index+1);
my $oldstring="$a $b $c";
my ($d,$e,$f) = solution ($n-$firstnum,$index+1);
$d*=$firstnum;
$e++;
$f++;
if ($a==$d) {
if ($e<$b) {
$b=$e;
}
}
if ($d>$a) {
$a=$d;
$b=$e;
}
if ($f>$c) {
$c=$f;
}
if ($a==$zero) {
$c=0;
$b=0;
}
#if ($n<$magic) {
if ($oldstring ne "$a $b $c") {
$a{$string}=[$a, $b, $c];
$hashsize++;
print "new entry for $n, $firstnum: $a $b $c\n";
if ($a>$zero && !defined $early[$n]) {
$early[$n]=$firstnum;
}
if (!defined $late[$n] && $firstnum>$max{$n}) {
$max{$n}=$firstnum;
}
}
#}
return ($a,$b,$c);
}
use Math::BigInt;
$zero= new Math::BigInt '+0';
$one= new Math::BigInt '+1';
$n=1;
@primes=();
$sumprimes=0;
print "Run started at: \n";
system ("date");
while (1) {
$n++;
$flag=0;
$max{$n}=0;
$m=int(sqrt($n));
for (reverse @primes) {
if ($_>$m) {
last;
}
if ($n%$_==0) {
$flag=1;
last;
}
}
if ($flag==0) {
unshift (@primes,$n);
$sumprimes+=$n;
$sumprimes{$n}=$sumprimes;
#print "$n is prime\n";
}
($a,$b,$c)=solution($n,0);
$late[$n]=$max{$n};
delete $max{$n};
if ($b!=$c) {
print "$n: $a $b $c (hashsize: $hashsize)\n";
#print OUT "$n $a $b $c\n";
system("date");
}
}
Here's a floating-point version, which is much faster. The answers it produces are logarithms of the products.
#use integer;
sub solution {
my $n=shift;
if ($n<2) {
return (($n==1)?undef:0,0,0);
}
if (($n==4) || ($n==6)) {
return (undef,0,0);
}
my $index=shift;
my $firstnum=$primes[$index];
#print "calling $n $firstnum\n";
if (defined $early[$n] && $firstnum<$early[$n]) {
return (undef,0,0);
}
if ($sumprimes{$firstnum}<$n) {
return (undef,0,0);
}
if (defined $late[$n]) {
if ($firstnum>$late[$n]) {
return @{$a{"$n $late[$n]"}};
} else {
while (!exists $a{"$n $firstnum"}) {
$firstnum=$primes[++$index]
}
return @{$a{"$n $firstnum"}}
}
}
#if ($n<$magic) {
my $string="$n $firstnum";
if (exists $a{$string}) {
return @{$a{$string}};
}
# }
my ($a,$b,$c) = solution ($n,$index+1);
my $oldstring="$a $b $c";
#if ignoring this prime is correct, don't remember info about it
my ($d,$e,$f) = solution ($n-$firstnum,$index+1);
if (!defined $a && !defined $d) {
return (undef,0,0);
}
if (defined $d) {
$d+=$l{$firstnum};
$e++;
$f++;
} else {
$e=0;
$f=0;
}
if (defined $a && defined $d && $a==$d) {
if ($e<$b) {
$b=$e;
}
}
if ($d>$a) {
$a=$d;
$b=$e;
}
if ($f>$c) {
$c=$f;
}
#if (!defined $a) {
# return (undef,0,0);
# }
#if ($n<$magic) {
if ($oldstring ne "$a $b $c") {
$a{$string}=[$a, $b, $c];
$hashsize++;
print "new entry for $n, $firstnum: $a $b $c\n";
if (defined $a && !defined $early[$n]) {
$early[$n]=$firstnum;
}
if (!defined $late[$n] && $firstnum>$max{$n}) {
$max{$n}=$firstnum;
}
}
#}
return ($a,$b,$c);
}
#use Math::BigInt;
#$zero= new Math::BigInt '+0';
#$one= new Math::BigInt '+1';
$filename=shift;
if (!defined $filename) {
$filename="distinctprimes.output";
}
chomp $filename;
open OUT, ">$filename";
$n=1;
$magic=1400;
@primes=();
$sumprimes=0;
print "Run started at: \n";
system ("date");
while ($n<25000) {
$n++;
$flag=0;
$max{$n}=0;
$m=int(sqrt($n));
for (reverse @primes) {
if ($_>$m) {
last;
}
if ($n%$_==0) {
$flag=1;
last;
}
}
if ($flag==0) {
unshift (@primes,$n);
$l{$n}=log($n);
$sumprimes+=$n;
$sumprimes{$n}=$sumprimes;
print "$n is prime\n";
}
($a,$b,$c)=solution($n,0);
$late[$n]=$max{$n};
delete $max{$n};
if ($b!=$c) {
print "$n: $a $b $c (hashsize: $hashsize)\n";
print OUT "$n $a $b $c\n";
system("date");
}
}
Here are the solutions I know so far. The format is number, largest prime used in best breakdown, natural log of best breadown, number of primes in best breakdown, number of primes in longest breakdown. The hashsize thing lets me know how many previous results the program remembers. I'm basically too lazy to edit it out.
319: 53 38.3211621017377 13 14 (hashsize: 378) 372: 59 42.2914540152898 14 15 (hashsize: 451) 492: 67 50.4798653233689 16 17 (hashsize: 618) 703: 79 63.2376972609495 19 20 (hashsize: 918) 865: 89 72.0259857212131 21 22 (hashsize: 1143) 954: 97 76.5146220909453 22 23 (hashsize: 1273) 1584: 127 104.430747109923 28 29 (hashsize: 2216) 1842: 137 114.150131519583 30 31 (hashsize: 2616) 2112: 149 123.385547170135 32 33 (hashsize: 3032) 2118: 149 124.004586378542 32 33 (hashsize: 3039) 2418: 157 134.025812521302 34 35 (hashsize: 3514) 2569: 163 138.463019118244 35 36 (hashsize: 3770) 2575: 163 139.08205832665 35 36 (hashsize: 3777) 2899: 173 148.674763131468 37 38 (hashsize: 4281) 2905: 173 149.293802339874 37 38 (hashsize: 4288) 3078: 179 154.447093934372 38 39 (hashsize: 4590) 3432: 191 164.213937563072 40 41 (hashsize: 5164) 3438: 191 164.832976771478 40 41 (hashsize: 5171) 4212: 211 185.305409733486 44 45 (hashsize: 6401) 4218: 211 185.924448941892 44 45 (hashsize: 6408) 4423: 223 190.657267866962 45 46 (hashsize: 6772) 4429: 223 191.276307075368 45 46 (hashsize: 6779) 5341: 239 212.99318932143 49 50 (hashsize: 8344) 5815: 251 223.335410598446 51 52 (hashsize: 9101) 5821: 251 223.954449806852 51 52 (hashsize: 9108) 6066: 257 228.860863537577 52 53 (hashsize: 9527) 6072: 257 229.479902745984 52 53 (hashsize: 9534) 6323: 263 234.409939622473 53 54 (hashsize: 9990) 6329: 263 235.028978830879 53 54 (hashsize: 9997) 6592: 269 240.601132863057 54 55 (hashsize: 10477) 7132: 277 251.797963063538 56 57 (hashsize: 11395) 7967: 293 268.086742928296 59 60 (hashsize: 12751) 7973: 293 268.705782136702 59 60 (hashsize: 12758) 8254: 307 273.387425915608 60 61 (hashsize: 13256) 8260: 307 273.766915537313 60 61 (hashsize: 13263) 8266: 307 274.38595474572 60 61 (hashsize: 13270) 9502: 331 296.359171539792 64 65 (hashsize: 15364) 9508: 331 296.738661161497 64 65 (hashsize: 15371) 9514: 331 297.357700369903 64 65 (hashsize: 15378) 9839: 337 302.540779536874 65 66 (hashsize: 15973) 9845: 337 303.15981874528 65 66 (hashsize: 15980) 10176: 347 308.360862467227 66 67 (hashsize: 16581) 10182: 347 308.979901675633 66 67 (hashsize: 16588) 11225: 359 325.931727226309 69 70 (hashsize: 18433) 11231: 359 326.550766434715 69 70 (hashsize: 18440) 11590: 367 332.434088823204 70 71 (hashsize: 19028) 11598: 367 332.45612828277 70 71 (hashsize: 19036) 11951: 373 337.720411462852 71 72 (hashsize: 19658) 11957: 373 338.339450671258 71 72 (hashsize: 19665) 12324: 379 343.641989882496 72 73 (hashsize: 20349) 12330: 379 344.261029090902 72 73 (hashsize: 20356) 13086: 389 355.527561076759 74 75 (hashsize: 21672) 13092: 389 356.146600285165 74 75 (hashsize: 21679) 13481: 397 362.110179628784 75 76 (hashsize: 22374) 13866: 401 367.09558707936 76 77 (hashsize: 23035) 13890: 401 368.124497993159 76 77 (hashsize: 23062) 14279: 409 374.088077336777 77 78 (hashsize: 23714) 14682: 419 379.482753284414 78 79 (hashsize: 24448) 14688: 419 380.10179249282 78 79 (hashsize: 24455) 15522: 431 391.563257038018 80 81 (hashsize: 25931) 15528: 431 392.182296246425 80 81 (hashsize: 25938) 16392: 439 404.319142064531 82 83 (hashsize: 27482) 17268: 449 415.878172039245 84 85 (hashsize: 28992) 17274: 449 416.497211247651 84 85 (hashsize: 28999) 17723: 457 422.604234135393 85 86 (hashsize: 29806) 19565: 479 446.527332672633 89 90 (hashsize: 33060) 19571: 479 447.146371881039 89 90 (hashsize: 33067) 20050: 487 453.31807247845 90 91 (hashsize: 33899) 21028: 499 465.702780729327 92 93 (hashsize: 35655) 22024: 509 477.516937786772 94 95 (hashsize: 37395) 22030: 509 478.135976995179 94 95 (hashsize: 37402) 22527: 521 483.369896181618 95 96 (hashsize: 38356) 22533: 521 483.749385803323 95 96 (hashsize: 38363) 22539: 521 484.368425011729 95 96 (hashsize: 38370) 23044: 523 489.434590986609 96 97 (hashsize: 39202) 23074: 523 490.651308500997 96 97 (hashsize: 39238) 23571: 541 495.885227687436 97 98 (hashsize: 40132) 23577: 541 496.264717309141 97 98 (hashsize: 40139) 23583: 541 496.883756517547 97 98 (hashsize: 40146) 24118: 547 502.558136587988 98 99 (hashsize: 41191) 24124: 547 503.177175796394 98 99 (hashsize: 41198) 24140: 557 503.206321757475 98 99 (hashsize: 41217) 24665: 557 508.86258539041 99 100 (hashsize: 42237) 24671: 557 509.481624598816 99 100 (hashsize: 42244) 24681: 557 509.499741036321 99 100 (hashsize: 42254) 25222: 563 515.185150630337 100 101 (hashsize: 43336) 25228: 563 515.804189838743 100 101 (hashsize: 43343) 25791: 569 522.137469466883 101 102 (hashsize: 44434) 26931: 577 534.828739110665 103 104 (hashsize: 46454) 27502: 587 540.567542168767 104 105 (hashsize: 47437) 27508: 587 541.186581377173 104 105 (hashsize: 47444) 27518: 587 541.203763930493 104 105 (hashsize: 47454) 28089: 593 546.942566988595 105 106 (hashsize: 48583) 28095: 593 547.561606197001 105 106 (hashsize: 48590) 28688: 599 553.946800595999 106 107 (hashsize: 49753) 29888: 607 566.74065712865 108 109 (hashsize: 51909) 30489: 613 572.530146711303 109 110 (hashsize: 53043) 30495: 613 573.149185919709 109 110 (hashsize: 53050) 32332: 631 591.421996322124 112 113 (hashsize: 56313) 32338: 631 591.801485943829 112 113 (hashsize: 56320) 32344: 631 592.420525152236 112 113 (hashsize: 56327) 32969: 641 598.248791806371 113 114 (hashsize: 57528) 32975: 641 598.867831014777 113 114 (hashsize: 57535) 34900: 653 617.65031228203 116 117 (hashsize: 61157) 34906: 653 618.269351490436 116 117 (hashsize: 61164) 35559: 659 624.750928619712 117 118 (hashsize: 62426) 36873: 673 637.11636678566 119 120 (hashsize: 64733) 36879: 673 637.735405994067 119 120 (hashsize: 64740) 38223: 683 650.145783388218 121 122 (hashsize: 67166) 38229: 683 650.764822596624 121 122 (hashsize: 67173) 38912: 691 657.291317456194 122 123 (hashsize: 68359) 38920: 691 657.302962420391 122 123 (hashsize: 68367) 39597: 701 663.210418071556 123 124 (hashsize: 69545) 39603: 701 663.829457279962 123 124 (hashsize: 69552) 39613: 701 663.843825343229 123 124 (hashsize: 69562) 39621: 701 663.855470307426 123 124 (hashsize: 69570) 40304: 709 670.381965166997 124 125 (hashsize: 70723) 40312: 709 670.393312806494 124 125 (hashsize: 70731) 40322: 709 670.407680869761 124 125 (hashsize: 70741) 41007: 719 676.326781485123 125 126 (hashsize: 72000) 41013: 719 676.945820693529 125 126 (hashsize: 72007) 41023: 719 676.959826524718 125 126 (hashsize: 72017) 41031: 719 676.971174164215 125 126 (hashsize: 72025) 41732: 727 683.52368205125 126 127 (hashsize: 73338) 41740: 727 683.534747171062 126 127 (hashsize: 73346) 42453: 733 689.493569320377 127 128 (hashsize: 74749) 42459: 733 690.112608528783 127 128 (hashsize: 74756) 43186: 739 696.090715022264 128 129 (hashsize: 76201) 43192: 739 696.70975423067 128 129 (hashsize: 76208) 44674: 751 709.925748196336 130 131 (hashsize: 78907) 45419: 757 715.928114639694 131 132 (hashsize: 80328) 45425: 757 716.5471538481 131 132 (hashsize: 80335) 46943: 769 729.811150459399 133 134 (hashsize: 83111) 48473: 787 742.107991647381 135 136 (hashsize: 85765) 48479: 787 742.487481269086 135 136 (hashsize: 85772) 48485: 787 743.106520477492 135 136 (hashsize: 85779) 49266: 797 749.155709517504 136 137 (hashsize: 87219) 49272: 797 749.77474872591 136 137 (hashsize: 87226) 49282: 797 749.787375156283 136 137 (hashsize: 87236) 50057: 809 755.457074574589 137 138 (hashsize: 88814) 50063: 809 755.836564196294 137 138 (hashsize: 88821) 50069: 809 756.4556034047 137 138 (hashsize: 88828) 51683: 821 769.230631167468 139 140 (hashsize: 91995) 51689: 821 769.849670375874 139 140 (hashsize: 92002) 54983: 839 796.092135327756 143 144 (hashsize: 97994) 54989: 839 796.711174536162 143 144 (hashsize: 98001) 55816: 853 802.444856412518 144 145 (hashsize: 99620) 55822: 853 802.824346034223 144 145 (hashsize: 99627) 55828: 853 803.443385242629 144 145 (hashsize: 99634) 59248: 877 829.463237491675 148 149 (hashsize: 105976) 59254: 877 829.84272711338 148 149 (hashsize: 105983) 59260: 877 830.461766321787 148 149 (hashsize: 105990) 62772: 907 856.400917056135 152 153 (hashsize: 112407) 62776: 907 856.591972292897 152 153 (hashsize: 112412) 62782: 907 856.971461914602 152 153 (hashsize: 112419) 62788: 907 857.590501123008 152 153 (hashsize: 112426) 63683: 911 863.402114743012 153 154 (hashsize: 114147) 63707: 919 864.413787245364 153 154 (hashsize: 114175) 64606: 919 871.215186470384 154 155 (hashsize: 115872) 65519: 929 877.419433384333 155 156 (hashsize: 117528) 65525: 929 878.038472592739 155 156 (hashsize: 117535) 65535: 929 878.049295209197 155 156 (hashsize: 117545) 66454: 937 884.872581331553 156 157 (hashsize: 119344) 68326: 947 897.943168544971 158 159 (hashsize: 122927) 68332: 947 898.562207753377 158 159 (hashsize: 122934) 69273: 953 904.796467638157 159 160 (hashsize: 124635) 69279: 953 905.415506846563 159 160 (hashsize: 124642) 70220: 967 911.276592920106 160 161 (hashsize: 126400) 70226: 967 911.656082541811 160 161 (hashsize: 126407) 70232: 967 912.275121750217 160 161 (hashsize: 126414) 72164: 977 925.408607505556 162 163 (hashsize: 130119) 72170: 977 926.027646713962 162 163 (hashsize: 130126) 73141: 983 932.293094157598 163 164 (hashsize: 131968) 73147: 983 932.912133366005 163 164 (hashsize: 131975) 74114: 991 938.613158419278 164 165 (hashsize: 133688) 74130: 991 939.802742486152 164 165 (hashsize: 133708) 74138: 991 939.810847900335 164 165 (hashsize: 133716) 75109: 997 945.702928190371 165 166 (hashsize: 135491) 75115: 997 946.082417812076 165 166 (hashsize: 135498) 75121: 997 946.701457020482 165 166 (hashsize: 135505) 76112: 1009 952.987168582037 166 167 (hashsize: 137511) 76118: 1009 953.606207790444 166 167 (hashsize: 137518) 78140: 1019 967.443594315046 168 169 (hashsize: 141597) 80174: 1031 980.679669958027 170 171 (hashsize: 145377) 80180: 1031 981.298709166433 170 171 (hashsize: 145384) 82244: 1039 995.17721611957 172 173 (hashsize: 149168) 83277: 1049 1001.50419090226 173 174 (hashsize: 151158) 83283: 1049 1002.12323011067 173 174 (hashsize: 151165) 85377: 1061 1015.41728088154 175 176 (hashsize: 155131) 85383: 1061 1016.03632008994 175 176 (hashsize: 155138) 87507: 1069 1029.9721376069 177 178 (hashsize: 158995) 88564: 1087 1035.94808768781 178 179 (hashsize: 161055) 88570: 1087 1036.32757730952 178 179 (hashsize: 161062) 88576: 1087 1036.94661651792 178 179 (hashsize: 161069) 92938: 1103 1064.31062013092 182 183 (hashsize: 169412) 92944: 1103 1064.92965933933 182 183 (hashsize: 169419) 94041: 1109 1071.31640915018 183 184 (hashsize: 171480) 94047: 1109 1071.93544835858 183 184 (hashsize: 171487) 95140: 1117 1077.75707827906 184 185 (hashsize: 173386) 95156: 1117 1078.94666234593 184 185 (hashsize: 173406) 95164: 1117 1078.95385015765 184 185 (hashsize: 173414) 96261: 1123 1084.96653531489 185 186 (hashsize: 175370) 96267: 1123 1085.3460249366 185 186 (hashsize: 175377) 96273: 1123 1085.965064145 185 186 (hashsize: 175384) 97384: 1129 1091.99029426963 186 187 (hashsize: 177311) 97390: 1129 1092.36978389133 186 187 (hashsize: 177318) 97396: 1129 1092.98882309974 186 187 (hashsize: 177325) 97416: 1129 1093.00669667654 186 187 (hashsize: 177345) 98509: 1151 1098.82832659702 187 188 (hashsize: 179446) 98513: 1151 1099.01938183378 187 188 (hashsize: 179451) 98519: 1151 1099.39887145548 187 188 (hashsize: 179458) 98525: 1151 1100.01791066389 187 188 (hashsize: 179465) 99688: 1163 1107.07666881641 188 189 (hashsize: 181793) 100823: 1163 1113.49738038448 189 190 (hashsize: 183960) 100829: 1163 1114.11641959288 189 190 (hashsize: 183967) 101992: 1171 1121.1751777454 190 191 (hashsize: 186172) 102000: 1171 1121.18203295648 190 191 (hashsize: 186180) 103157: 1181 1127.62175190059 191 192 (hashsize: 188463) 103163: 1181 1128.240791109 191 192 (hashsize: 188470) 103173: 1181 1128.2492945616 191 192 (hashsize: 188480) 104338: 1187 1134.69586871679 192 193 (hashsize: 190841) 104344: 1187 1135.3149079252 192 193 (hashsize: 190848) 105519: 1193 1141.39556348969 193 194 (hashsize: 193169) 105525: 1193 1141.7750531114 193 194 (hashsize: 193176) 105531: 1193 1142.39409231981 193 194 (hashsize: 193183) 106708: 1201 1148.28873467503 194 195 (hashsize: 195439) 106724: 1201 1149.4783187419 194 195 (hashsize: 195459) 106732: 1201 1149.48500214189 194 195 (hashsize: 195467) 107919: 1213 1155.95018935558 195 196 (hashsize: 197851) 107925: 1213 1156.56922856398 195 196 (hashsize: 197858) 110349: 1223 1170.15518535751 197 198 (hashsize: 202684) 110355: 1223 1170.77422456591 197 198 (hashsize: 202691) 111578: 1229 1177.8832867016 198 199 (hashsize: 205144) 114038: 1237 1192.11282493735 200 201 (hashsize: 209557) 115263: 1249 1198.23474047963 201 202 (hashsize: 211787) 115269: 1249 1198.61423010134 201 202 (hashsize: 211794) 115275: 1249 1199.23326930975 201 202 (hashsize: 211801) 115287: 1249 1199.24292344748 201 202 (hashsize: 211813) 116508: 1259 1205.173783753 202 203 (hashsize: 213950) 116518: 1259 1205.74432861146 202 203 (hashsize: 213963) 116524: 1259 1206.36336781987 202 203 (hashsize: 213970) 116534: 1259 1206.37134234379 202 203 (hashsize: 213980) 116546: 1259 1206.38099648152 202 203 (hashsize: 213992) 117771: 1277 1212.5029120238 203 204 (hashsize: 216571) 117777: 1277 1212.88240164551 203 204 (hashsize: 216578) 117783: 1277 1213.50144085392 203 204 (hashsize: 216585) 121622: 1289 1234.96449987614 206 207 (hashsize: 224391) 124202: 1297 1249.28929426993 208 209 (hashsize: 229163) 129404: 1319 1277.3568670151 212 213 (hashsize: 238800) 129410: 1319 1277.97590622351 212 213 (hashsize: 238807) 132050: 1327 1292.34667968075 214 215 (hashsize: 243477) 133353: 1361 1298.04927865965 215 216 (hashsize: 245830) 133361: 1361 1298.3477716482 215 216 (hashsize: 245840) 133365: 1361 1298.53882688497 215 216 (hashsize: 245845) 133371: 1361 1298.91831650667 215 216 (hashsize: 245852) 133377: 1361 1299.53735571508 215 216 (hashsize: 245859) 133411: 1361 1299.5626546834 215 216 (hashsize: 245895) 134726: 1367 1305.75480188762 216 217 (hashsize: 248652) 134732: 1367 1306.13429150932 216 217 (hashsize: 248659) 134738: 1367 1306.75333071773 216 217 (hashsize: 248666) 134758: 1381 1306.76791886849 216 217 (hashsize: 248693) 136093: 1373 1312.97517572434 217 218 (hashsize: 251423) 136099: 1373 1313.35466534605 217 218 (hashsize: 251430) 136105: 1373 1313.97370455445 217 218 (hashsize: 251437) 137462: 1381 1320.00887389335 218 219 (hashsize: 254127) 137478: 1381 1321.19845796022 218 219 (hashsize: 254147) 137486: 1381 1321.20426770786 218 219 (hashsize: 254155) 138847: 1399 1327.43049228352 219 220 (hashsize: 256745) 138853: 1399 1327.80998190522 219 220 (hashsize: 256752) 138859: 1399 1328.42902111363 219 220 (hashsize: 256759) 138877: 1399 1328.44197093489 219 220 (hashsize: 256777) 140242: 1409 1334.48295002142 220 221 (hashsize: 259582) 140252: 1409 1335.05349487989 220 221 (hashsize: 259595) 140258: 1409 1335.6725340883 220 221 (hashsize: 259602) 140268: 1409 1335.67965662553 220 221 (hashsize: 259612) 141655: 1423 1341.92464077008 221 222 (hashsize: 262641) 141661: 1423 1342.30413039179 221 222 (hashsize: 262648) 141667: 1423 1342.9231696002 221 222 (hashsize: 262655) 147373: 1439 1371.36023821311 225 226 (hashsize: 273936) 147379: 1439 1371.97927742152 225 226 (hashsize: 273943) 148818: 1447 1379.25098112841 226 227 (hashsize: 276664) 150253: 1451 1385.52970002493 227 228 (hashsize: 279525) 153169: 1459 1401.08962277149 229 230 (hashsize: 284816) 154616: 1471 1407.3766004899 230 231 (hashsize: 287496) 154622: 1471 1407.75609011161 230 231 (hashsize: 287503) 154628: 1471 1408.37512932002 230 231 (hashsize: 287510) 154640: 1471 1408.38332049209 230 231 (hashsize: 287522) 156093: 1481 1415.04978783221 231 232 (hashsize: 290572) 156099: 1481 1415.66882704062 231 232 (hashsize: 290579) 163526: 1499 1451.5710017653 236 237 (hashsize: 304762) 163532: 1499 1452.19004097371 236 237 (hashsize: 304769) 165019: 1511 1458.5040656417 237 238 (hashsize: 307427) 165025: 1511 1458.8835552634 237 238 (hashsize: 307434) 165031: 1511 1459.50259447181 237 238 (hashsize: 307441) 165043: 1511 1459.51056793598 237 238 (hashsize: 307453) 166530: 1523 1465.82459260397 238 239 (hashsize: 310395) 166536: 1523 1466.20408222568 238 239 (hashsize: 310402) 166542: 1523 1466.82312143408 238 239 (hashsize: 310409) 166554: 1523 1466.8310318247 238 239 (hashsize: 310421) 168049: 1531 1472.9619747201 239 240 (hashsize: 313371) 168065: 1531 1474.15155878698 239 240 (hashsize: 313391) 168073: 1531 1474.15679782974 239 240 (hashsize: 313399) 168085: 1531 1474.16470822036 239 240 (hashsize: 313411) 169584: 1543 1480.48670635252 240 241 (hashsize: 316506) 169590: 1543 1480.86619597423 240 241 (hashsize: 316513) 169596: 1543 1481.48523518263 240 241 (hashsize: 316520) 169608: 1543 1481.49304263934 240 241 (hashsize: 316532) 171133: 1549 1488.20767982659 241 242 (hashsize: 319831) 171139: 1549 1488.826719035 241 242 (hashsize: 319838) 174235: 1559 1502.90098849016 243 244 (hashsize: 326033) 174241: 1559 1503.52002769856 243 244 (hashsize: 326040) 175800: 1567 1510.87182756762 244 245 (hashsize: 329115) 178938: 1579 1525.58821344823 246 247 (hashsize: 335157) 182088: 1597 1539.32130869226 248 249 (hashsize: 341155) 182094: 1597 1539.70079831396 248 249 (hashsize: 341162) 182100: 1597 1540.31983752237 248 249 (hashsize: 341169) 185298: 1607 1555.07410338358 250 251 (hashsize: 347914) 190127: 1619 1577.22544697444 253 254 (hashsize: 357340) 193367: 1627 1592.00580944985 255 256 (hashsize: 363147) 194988: 1637 1598.78126334866 256 257 (hashsize: 365930) 194994: 1637 1599.40030255707 256 257 (hashsize: 365937) 195004: 1637 1599.40643002722 256 257 (hashsize: 365947) 196615: 1657 1605.61133906757 257 258 (hashsize: 369122) 196619: 1657 1605.80239430433 257 258 (hashsize: 369127) 196625: 1657 1606.18188392603 257 258 (hashsize: 369134) 196631: 1657 1606.80092313444 257 258 (hashsize: 369141) 196645: 1657 1606.19402736609 257 258 (hashsize: 369155) 196651: 1657 1606.81306657449 257 258 (hashsize: 369161) 198282: 1663 1613.59464794346 258 259 (hashsize: 372585) 198288: 1663 1614.21368715187 258 259 (hashsize: 372592) 203271: 1693 1635.2792423706 261 262 (hashsize: 382229) 203275: 1693 1635.47029760736 261 262 (hashsize: 382234) 203281: 1693 1635.84978722906 261 262 (hashsize: 382241) 203287: 1693 1636.46882643747 261 262 (hashsize: 382248) 204996: 1709 1643.91249012059 262 263 (hashsize: 385683) 208370: 1709 1658.1584569981 264 265 (hashsize: 392295) 208376: 1709 1658.77749620651 264 265 (hashsize: 392302) 210073: 1721 1665.22263105951 265 266 (hashsize: 395683) 210079: 1721 1665.60212068122 265 266 (hashsize: 395690) 210085: 1721 1666.22115988963 265 266 (hashsize: 395697) 213523: 1733 1680.50460371396 267 268 (hashsize: 402400) 213529: 1733 1681.12364292237 267 268 (hashsize: 402407) 215262: 1741 1688.58125221208 268 269 (hashsize: 405780) 215270: 1741 1688.58585786213 268 269 (hashsize: 405788) 216991: 1747 1695.04493832174 269 270 (hashsize: 409168) 216997: 1747 1695.42442794344 269 270 (hashsize: 409175) 217003: 1747 1696.04346715185 269 270 (hashsize: 409182) 218738: 1753 1702.51059363187 270 271 (hashsize: 412426) 218744: 1753 1702.89008325358 270 271 (hashsize: 412433) 218750: 1753 1703.50912246198 270 271 (hashsize: 412440) 220491: 1759 1709.97967751679 271 272 (hashsize: 415704) 220497: 1759 1710.3591671385 271 272 (hashsize: 415711) 220503: 1759 1710.9782063469 271 272 (hashsize: 415718) 222250: 1777 1717.45217826153 272 273 (hashsize: 419412) 222256: 1777 1717.83166788324 272 273 (hashsize: 419419) 222262: 1777 1718.45070709164 272 273 (hashsize: 419426) 222274: 1777 1717.84184896665 272 273 (hashsize: 419439) 222280: 1777 1718.46088817506 272 273 (hashsize: 419445) 224033: 1783 1725.31434971139 273 274 (hashsize: 423196) 224039: 1783 1725.9333889198 273 274 (hashsize: 423203) 229386: 1801 1747.39861830622 276 277 (hashsize: 433587) 229392: 1801 1747.77810792792 276 277 (hashsize: 433594) 229398: 1801 1748.39714713633 276 277 (hashsize: 433601) 231183: 1811 1754.70366041463 277 278 (hashsize: 436893) 231193: 1811 1755.2742052731 277 278 (hashsize: 436906) 231199: 1811 1755.8932444815 277 278 (hashsize: 436913) 231209: 1811 1755.89878159421 277 278 (hashsize: 436923) 232998: 1823 1762.39635010928 278 279 (hashsize: 440391) 233004: 1823 1762.77583973098 278 279 (hashsize: 440398) 233010: 1823 1763.39487893939 278 279 (hashsize: 440405) 233022: 1823 1763.40148325618 278 279 (hashsize: 440417) 234817: 1831 1769.71353364719 279 280 (hashsize: 443891) 234833: 1831 1770.90311771407 279 280 (hashsize: 443911) 234841: 1831 1770.90749648406 279 280 (hashsize: 443919) 236634: 1847 1776.75072749515 280 281 (hashsize: 447511) 236640: 1847 1776.92765820331 280 281 (hashsize: 447520) 236658: 1847 1777.79669605033 280 281 (hashsize: 447543) 236664: 1847 1778.41573525874 280 281 (hashsize: 447550) 236680: 1847 1778.42443569427 280 281 (hashsize: 447566) 236688: 1847 1778.42881446426 280 281 (hashsize: 447574) 238495: 1861 1784.74746917207 281 282 (hashsize: 451563) 238499: 1861 1784.93852440883 281 282 (hashsize: 451568) 238505: 1861 1785.31801403053 281 282 (hashsize: 451575) 238511: 1861 1785.93705323894 281 282 (hashsize: 451582) 238519: 1861 1785.32556530698 281 282 (hashsize: 451590) 238525: 1861 1785.94460451538 281 282 (hashsize: 451596) 240366: 1867 1792.84688328718 282 283 (hashsize: 455905) 240372: 1867 1793.46592249558 282 283 (hashsize: 455912) 249733: 1889 1830.52442149544 287 288 (hashsize: 473942) 249739: 1889 1831.14346070384 287 288 (hashsize: 473949) 251616: 1901 1837.68873474123 288 289 (hashsize: 477365) 251622: 1901 1838.06822436294 288 289 (hashsize: 477372) 251628: 1901 1838.68726357134 288 289 (hashsize: 477379) 251640: 1901 1838.69359604633 288 289 (hashsize: 477391) 253517: 1907 1845.23887008372 289 290 (hashsize: 480992) 253523: 1907 1845.61835970543 289 290 (hashsize: 480999) 253529: 1907 1846.23739891383 289 290 (hashsize: 481006) 255424: 1913 1852.79215668932 290 291 (hashsize: 484758) 255430: 1913 1853.17164631103 290 291 (hashsize: 484765) 255436: 1913 1853.79068551943 290 291 (hashsize: 484772) 257337: 1931 1860.34858465876 291 292 (hashsize: 488749) 257343: 1931 1860.72807428047 291 292 (hashsize: 488756) 257349: 1931 1861.34711348887 291 292 (hashsize: 488763) 261207: 1949 1875.8606960421 293 294 (hashsize: 496268) 261213: 1949 1876.47973525051 293 294 (hashsize: 496275) 265097: 1973 1890.44132022377 295 296 (hashsize: 503735) 265101: 1973 1890.63237546053 295 296 (hashsize: 503740) 265107: 1973 1891.01186508223 295 296 (hashsize: 503747) 265113: 1973 1891.63090429064 295 296 (hashsize: 503754) 267074: 1979 1898.21968596655 296 297 (hashsize: 508074) 267080: 1979 1898.59917558826 296 297 (hashsize: 508081) 267086: 1979 1899.21821479666 296 297 (hashsize: 508088) 267100: 1987 1899.22528553319 296 297 (hashsize: 508104) 269065: 1987 1906.80856174227 297 298 (hashsize: 512361) 269073: 1987 1906.81259603921 297 298 (hashsize: 512369) 271046: 1993 1913.78390377641 298 299 (hashsize: 516725) 271052: 1993 1914.40294298482 298 299 (hashsize: 516732) 279044: 2011 1944.80254430861 302 303 (hashsize: 532119) 281049: 2017 1951.78989248998 303 304 (hashsize: 535924) 281055: 2017 1952.40893169838 303 304 (hashsize: 535931) 283066: 2027 1959.39925902793 304 305 (hashsize: 539988) 283072: 2027 1960.01829823634 304 305 (hashsize: 539995) 285111: 2029 1967.63854218466 305 306 (hashsize: 543773) 287122: 2039 1974.62886951421 306 307 (hashsize: 547496) 287128: 2039 1975.24790872262 306 307 (hashsize: 547503) 289151: 2053 1981.67853942632 307 308 (hashsize: 551355) 289155: 2053 1981.86959466308 307 308 (hashsize: 551360) 289161: 2053 1982.24908428478 307 308 (hashsize: 551367) 289167: 2053 1982.86812349319 307 308 (hashsize: 551374) 289175: 2053 1982.25592693123 307 308 (hashsize: 551382) 289181: 2053 1982.87496613964 307 308 (hashsize: 551388) 291214: 2063 1989.8761417018 308 309 (hashsize: 555481) 291220: 2063 1990.49518091021 308 309 (hashsize: 555488) 291230: 2063 1990.50004000626 308 309 (hashsize: 555498) 293277: 2069 1997.50805821488 309 310 (hashsize: 559862) 293283: 2069 1998.12709742328 309 310 (hashsize: 559869) 295340: 2081 2004.76338927092 310 311 (hashsize: 564300) 295346: 2081 2005.14287889262 310 311 (hashsize: 564307) 295352: 2081 2005.76191810103 310 311 (hashsize: 564314) 303686: 2099 2035.71297082891 314 315 (hashsize: 580836) 303692: 2099 2036.33201003732 314 315 (hashsize: 580843) 305779: 2111 2042.98269752702 315 316 (hashsize: 584966) 305785: 2111 2043.36218714873 315 316 (hashsize: 584973) 305791: 2111 2043.98122635714 315 316 (hashsize: 584980)
I'm not a number theorist, so I don't really know how interesting this sequence is, or how hard it is to say things about it. Is there an easier way of generating terms in it? Silvio Sergio has a program which is much faster than mine (up to n=500000 in 3 minutes) but generates some false positives. Does the sequence go on forever? I would imagine so. Do numbers in it become rarer or more common? They seem to become more common in the limited sample I have here. Also, are there cases where the best breakdown has 2 fewer terms than the longest? k fewer, for general k?
I would be delighted to hear from anyone with information about this, especially if the problem is trivial and the numbers can be characterised in some easy way.