This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331479 #10 Jan 25 2020 18:21:01 %S A331479 1,1,1,2,1,1,2,1,1,2,1,1,2,1,3,1,2,4,1,1,2,1,1,2,4,1,3,1,2,4,5,1,3,1, %T A331479 2,4,1,3,1,2,4,1,7,1,2,1,3,1,2,4,5,1,3,1,2,4,1,3,1,2,4,5,8,1,3,9,1,2, %U A331479 4,5,8,1,3,1,2,4,7,1,3,1,2,4,1,3,1,2,4,8 %N A331479 Table read by rows: row n lists the numbers m such that the first n primes can be partitioned into m subsets all of which have the same sum. %C A331479 Consider the following one-dimensional bin packing problem: given n items whose sizes are the first n primes, list the numbers m such that all the items can be packed into m bins of identical capacity, with each bin packed completely full. The resulting list is row n. %C A331479 If a row contains a number m, it necessarily contains all divisors of m. %e A331479 In bin-packing terms, for n=19, the sum of the 19 item sizes, i.e., the sum of the first n primes, is 2 + 3 + ... + 67 = 568, whose divisors begin 1, 2, 4, 8, ...; the bin capacity must be at least 67 (the size of the largest item), and 568/67 < 9, so the number of bins m cannot exceed 8. However, the 19 items cannot be packed into 8 bins: the bin capacity would be 568/8 = 71 (which, as an odd sum, would require that each bin containing only odd-sized items -- i.e., every bin other than the one containing the item of size 2 -- contain an odd number of items, hence at least 3 items, but there are only 19 items in total). So the remaining values of m are 1 (i.e., packing all 19 items in a single bin), 2 (e.g., 568/2 = 284 = 67 + 61 + 59 + 53 + 41 + 3 = 47 + 43 + 37 + 31 + 29 + 23 + 19 + 17 + 13 + 11 + 7 + 5 + 2), and 4 (e.g., 568/4 = 142 = 67 + 61 + 11 + 3 = 59 + 53 + 23 + 7 = 47 + 43 + 37 + 13 + 2 = 41 + 31 + 29 + 19 + 17 + 5), so row 19 consists of the numbers 1, 2, and 4. %e A331479 . Numbers m such that %e A331479 Sum of Divisors m of sum 1st n primes can be %e A331479 n-th 1st n such that partitioned into m %e A331479 n prime primes m <= sum/prime(n) subsets w/same sum %e A331479 -- ----- ------ ----------------- ------------------- %e A331479 1 2 2 1 1; %e A331479 2 3 5 1 1; %e A331479 3 5 10 1, 2 1, 2; %e A331479 4 7 17 1 1; %e A331479 5 11 28 1, 2 1, 2; %e A331479 6 13 41 1 1; %e A331479 7 17 58 1, 2 1, 2; %e A331479 8 19 77 1 1; %e A331479 9 23 100 1, 2, 4 1, 2; %e A331479 10 29 129 1, 3 1, 3; %e A331479 11 31 160 1, 2, 4, 5 1, 2, 4; %e A331479 12 37 197 1 1; %e A331479 13 41 238 1, 2 1, 2; %e A331479 14 43 281 1 1; %e A331479 15 47 328 1, 2, 4 1, 2, 4; %e A331479 16 53 381 1, 3 1, 3; %e A331479 17 59 440 1, 2, 4, 5 1, 2, 4, 5; %e A331479 18 61 501 1, 3 1, 3; %e A331479 19 67 568 1, 2, 4, 8 1, 2, 4; %e A331479 20 71 639 1, 3, 9 1, 3; %e A331479 21 73 712 1, 2, 4, 8 1, 2, 4; %e A331479 22 79 791 1, 7 1, 7; %e A331479 23 83 874 1, 2 1, 2; %e A331479 24 89 963 1, 3, 9 1, 3; %e A331479 25 97 1060 1, 2, 4, 5, 10 1, 2, 4, 5; %e A331479 26 101 1161 1, 3, 9 1, 3; %e A331479 27 103 1264 1, 2, 4, 8 1, 2, 4; %e A331479 28 107 1371 1, 3 1, 3; %e A331479 29 109 1480 1, 2, 4, 5, 8, 10 1, 2, 4, 5, 8; %e A331479 30 113 1593 1, 3, 9 1, 3, 9; %e A331479 31 127 1720 1, 2, 4, 5, 8, 10 1, 2, 4, 5, 8; %e A331479 32 131 1851 1, 3 1, 3; %e A331479 33 137 1988 1, 2, 4, 7, 14 1, 2, 4, 7; %e A331479 34 139 2127 1, 3 1, 3; %e A331479 35 149 2276 1, 2, 4 1, 2, 4; %e A331479 36 151 2427 1, 3 1, 3; %e A331479 37 157 2584 1, 2, 4, 8 1, 2, 4, 8; %Y A331479 Cf. A000040, A007504, A053050, A306443, A327449. %K A331479 nonn,tabf %O A331479 1,4 %A A331479 _Jon E. Schoenfield_, Jan 17 2020