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 A279412 #8 Feb 16 2025 08:33:37
%S A279412 0,2,3,0,10,0,14,8,9,20,11,24,26,28,30,48,34,72,57,80,84,88,115,120,
%T A279412 125,156,135,168,203,180,279,224,297,306,315,396,407,418,507,480,574,
%U A279412 630,645,748,720,828,893,960,1029,1150,1122,1300,1378,1458,1650,1624,1824,1856,2065,2220,2379,2480,2646,2816,2925
%N A279412 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
%C A279412 Sum of all parts of all partitions of n into distinct primes.
%H A279412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H A279412 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A279412 G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
%F A279412 G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^prime(k)).
%F A279412 a(n) = n*A000586(n).
%e A279412 a(12) = 24 because we have [7, 5], [7, 3, 2] and 2*12 = 24.
%t A279412 nmax = 65; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}] Product[1 + x^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]]
%t A279412 nmax = 65; Rest[CoefficientList[Series[x D[Product[1 + x^Prime[k], {k, 1, nmax}], x], {x, 0, nmax}], x]]
%Y A279412 Cf. A000586, A024938, A066189, A276560.
%K A279412 nonn
%O A279412 1,2
%A A279412 _Ilya Gutkovskiy_, Apr 11 2017