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 A331387 #15 Apr 17 2021 14:13:45
%S A331387 1,2,4,7,11,16,24,34,47,64,86,113,148,191,245,310,390,486,602,740,907,
%T A331387 1104,1338,1613,1937,2315,2758,3272,3871,4562,5362,6283,7344,8558,
%U A331387 9952,11542,13356,15419,17766,20425,23440,26846,30696,35032,39917,45406
%N A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.
%C A331387 Primes of parts means the prime counting function applied to the part sizes. Equivalently, a(n) is the number of integer partitions with part sizes in A014689(n) interpreted as a multiset. - _Andrew Howroyd_, Apr 17 2021
%H A331387 Andrew Howroyd, <a href="/A331387/b331387.txt">Table of n, a(n) for n = 0..1000</a>
%F A331387 G.f.: 1/Product_{k>=1} 1 - x^(prime(k)-k). - _Andrew Howroyd_, Apr 16 2021
%e A331387 The a(0) = 1 through a(5) = 16 partitions:
%e A331387 () (1) (3) (4) (33) (43)
%e A331387 (2) (11) (31) (41) (331)
%e A331387 (21) (32) (42) (332)
%e A331387 (22) (111) (311) (411)
%e A331387 (211) (321) (421)
%e A331387 (221) (322) (422)
%e A331387 (222) (1111) (3111)
%e A331387 (2111) (3211)
%e A331387 (2211) (3221)
%e A331387 (2221) (3222)
%e A331387 (2222) (11111)
%e A331387 (21111)
%e A331387 (22111)
%e A331387 (22211)
%e A331387 (22221)
%e A331387 (22222)
%e A331387 For example, the partition (3,2,2,1) is counted under n = 5 because it has sum of primes 5+3+3+2 = 13 and its sum of parts plus n is also 3+2+2+1+5 = 13.
%t A331387 Table[Sum[Length[Select[IntegerPartitions[k],Total[Prime/@#]==k+n&]],{k,0,2*n}],{n,0,10}]
%o A331387 (PARI) seq(n)={my(m=1); while(prime(m)-m<=n, m++); Vec(1/prod(k=1, m, 1 - x^(prime(k)-k) + O(x*x^n)))} \\ _Andrew Howroyd_, Apr 16 2021
%Y A331387 Column sums of A331385.
%Y A331387 Partitions into primes are A000607.
%Y A331387 Partitions whose sum of primes is divisible by their sum are A331379.
%Y A331387 Partitions whose product divides their sum of primes are A331381.
%Y A331387 Partitions whose product equals their sum of primes are A331383.
%Y A331387 Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415, A331416, A331418.
%K A331387 nonn
%O A331387 0,2
%A A331387 _Gus Wiseman_, Jan 17 2020
%E A331387 Terms a(31) and beyond from _Andrew Howroyd_, Apr 16 2021