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 A276560 #14 Feb 16 2025 08:33:36
%S A276560 0,2,3,4,10,12,21,24,36,50,66,84,117,140,180,224,289,342,437,520,630,
%T A276560 770,920,1104,1300,1560,1809,2156,2523,2940,3441,3968,4620,5338,6125,
%U A276560 7092,8103,9272,10608,12080,13776,15624,17759,20064,22680,25622,28858,32496,36456,40950,45849,51324,57399,64044,71390
%N A276560 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).
%C A276560 Sum of all parts of all partitions of n into prime parts.
%C A276560 Convolution of the sequences A000607 and A008472.
%H A276560 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H A276560 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A276560 G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).
%F A276560 G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^prime(k)).
%F A276560 a(n) = n*A000607(n).
%F A276560 a(n) ~ n*exp(2*Pi*sqrt(n/log(n))/sqrt(3)).
%e A276560 a(6) = 12 because we have [3, 3], [2, 2, 2] and 2*6 = 12.
%t A276560 nmax = 55; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}] Product[1/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A276560 nmax = 55; Rest[CoefficientList[Series[x D[Product[1/(1 - x^Prime[k]), {k, 1, nmax}], x], {x, 0, nmax}], x]]
%t A276560 Table[Total@Flatten[IntegerPartitions[n,All,Prime@Range@PrimePi@n]],{n,52}] (* _Giorgos Kalogeropoulos_, Sep 12 2021 *)
%Y A276560 Cf. A000607, A008472, A066186, A084993.
%K A276560 nonn
%O A276560 1,2
%A A276560 _Ilya Gutkovskiy_, Apr 10 2017