A282289 - OEIS

%I #9 Feb 16 2025 08:33:41

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,4,4,0,0,6,12,6,0,8,12,12,4,

%T 13,16,6,4,13,28,12,4,10,24,24,16,28,24,24,24,42,52,18,28,32,60,40,24,

%U 44,28,42,28,60,52,18,24,37,84,54,48,42,60,78,48,72,44,60,52,68,96,36,40,22,72,72,52,76,52,66,36,88,88,64,56

%N Expansion of (Sum_{p prime, k>=2} x^(p^k))^4.

%C Number of ways to write n as an ordered sum of 4 proper prime powers (A246547).

%C Conjecture: a(n) > 0 for all n > 27.

%H Ilya Gutkovskiy, <a href="/A282289/a282289.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

%F G.f.: (Sum_{p prime, k>=2} x^(p^k))^4.

%e a(28) = 8 because we have [16, 4, 4, 4], [8, 8, 8, 4], [8, 8, 4, 8], [8, 4, 8, 8], [4, 16, 4, 4], [4, 8, 8, 8], [4, 4, 16, 4] and [4, 4, 4, 16].

%t nmax = 91; CoefficientList[Series[Sum[Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]^4, {x, 0, nmax}], x]

%Y Cf. A246547, A280242, A280243, A282062, A282064.

%K nonn,changed

%O 0,21

%A _Ilya Gutkovskiy_, Feb 11 2017