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A282062 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.

%I A282062 #12 Feb 16 2025 08:33:40
%S A282062 0,0,1,2,3,4,5,4,5,6,7,6,8,6,7,6,7,6,9,6,10,8,7,4,10,6,9,8,10,6,12,6,
%T A282062 13,10,13,8,14,4,11,8,12,6,12,6,12,10,11,4,16,6,15,8,12,4,17,6,14,8,
%U A282062 11,4,16,6,13,8,13,6,18,4,16,10,14,4,20,6,15,12,14,6,18,4,18,8,13,8,22,6,17,8,14,6,24,8,16,6,13,4
%N A282062 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.
%C A282062 Number of ways to write n as an ordered sum of two prime powers (1 included).
%H A282062 Robert Israel, <a href="/A282062/b282062.txt">Table of n, a(n) for n = 0..10000</a>
%H A282062 Ilya Gutkovskiy, <a href="/A282062/a282062.pdf">Extended graphical example</a>
%H A282062 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%F A282062 G.f.: (x + Sum_{p prime, k>=1} x^(p^k))^2.
%e A282062 a(8) = 5 because we have  [7, 1], [5, 3], [4, 4], [3, 5] and [1, 7].
%p A282062 N:= 100: # to get a(0)..a(N)
%p A282062 P:= select(isprime, [$2..N]):
%p A282062 g:= x + add(add(x^(p^k),k=1..floor(log[p](N))),p=P):
%p A282062 S:= series(g^2,x,N+1):
%p A282062 seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Feb 10 2017
%t A282062 nmax = 95; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]
%Y A282062 Cf. A000961, A071330, A071331, A073610, A095840, A280242, A282064.
%K A282062 nonn,look
%O A282062 0,4
%A A282062 _Ilya Gutkovskiy_, Feb 05 2017