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A106244 Number of partitions into distinct prime powers.

%I A106244 #30 Oct 30 2023 07:49:14
%S A106244 1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,14,17,19,21,24,27,30,33,37,41,46,50,
%T A106244 56,62,68,75,82,91,99,108,118,129,141,152,166,180,196,211,229,248,267,
%U A106244 288,310,335,360,387,415,447,479,513,549,589,630,672,719,768,820,873,930
%N A106244 Number of partitions into distinct prime powers.
%C A106244 A054685(n) < a(n) < A023893(n) for n>2.
%H A106244 Reinhard Zumkeller, <a href="/A106244/b106244.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%F A106244 a(n) = A054685(n-1)+A054685(n). - _Vladeta Jovovic_, Apr 28 2005
%F A106244 G.f.: (1+x)*Product(Product(1+x^(p(k)^j), j=1..infinity),k=1..infinity), where p(k) is the k-th prime (offset 0). - _Emeric Deutsch_, Aug 27 2007
%e A106244 a(10) = #{3^2+1,2^3+2,7+3,7+2+1,5+2^2+1,5+3+2,2^2+3+2+1} = 7.
%p A106244 g:=(1+x)*(product(product(1+x^(ithprime(k)^j),j=1..5),k=1..20)): gser:=series(g,x=0,68): seq(coeff(gser,x,n),n=1..63); # _Emeric Deutsch_, Aug 27 2007
%t A106244 m = 64; gf = (1+x)*Product[1+x^(Prime[k]^j), {j, 1, 5}, {k, 1, 18}] + O[x]^m; CoefficientList[gf, x] (* _Jean-François Alcover_, Mar 02 2019, from Maple *)
%o A106244 (PARI) lista(m) = {x = t + t*O(t^m); gf = (1+x)*prod(k=1, m, if (isprimepower(k),(1+x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ _Michel Marcus_, Mar 02 2019
%o A106244 (Haskell)
%o A106244 import Data.MemoCombinators (memo2, integral)
%o A106244 a106244 n = a106244_list !! n
%o A106244 a106244_list = map (p' 1) [0..] where
%o A106244    p' = memo2 integral integral p
%o A106244    p _ 0 = 1
%o A106244    p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
%o A106244            where pp = a000961 k
%o A106244 -- _Reinhard Zumkeller_, Nov 24 2015
%Y A106244 Cf. A000586, A000607, A000961.
%Y A106244 Cf. A062051, A105420, A131996.
%Y A106244 Cf. A023893, A051613, A054685.
%K A106244 nonn
%O A106244 0,4
%A A106244 _Reinhard Zumkeller_, Apr 26 2005
%E A106244 Offset corrected and a(0)=1 added by _Reinhard Zumkeller_, Nov 24 2015