A280596 - OEIS

%I #5 Feb 16 2025 08:33:39

%S 1,0,0,0,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,2,0,1,1,2,0,1,1,2,

%T 1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,3,1,1,3,3,1,1,3,3,2,1,3,4,2,1,3,4,2,1,

%U 3,4,2,1,3,4,3,1,4,4,3,1,4,5,3,2,4,6,3,2,4,6,4,2,4,6,4,2,4,6,4,2,4,7,4,2,4,7,5,2

%N Expansion of Product_{p prime, k>=2} (1 + x^(p^k)).

%C Number of partitions of n into distinct proper prime powers (A246547).

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

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{p prime, k>=2} (1 + x^(p^k)).

%e a(25) = 2 because we have [25] and [16, 9].

%t nmax = 107; CoefficientList[Series[Product[(1 + Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

%Y Cf. A054685, A106244, A246547, A280586.

%K nonn

%O 0,26

%A _Ilya Gutkovskiy_, Jan 06 2017