A339221 - OEIS

%I #9 May 03 2021 12:50:41

%S 1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,2,3,2,3,3,3,3,3,3,4,3,

%T 4,4,4,4,4,4,5,4,5,5,5,5,5,5,6,5,6,6,6,6,7,6,8,7,8,8,8,8,8,8,9,8,9,9,

%U 9,9,10,9,11,10,11,11,11,11,12,11,13,13,13,14,14,14,15,14,16,15,16,16,16,16,17,16

%N Number of partitions of n into cubes where every part appears at least 2 times.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: Product_{k>=1} (1 + x^(2*k^3) / (1 - x^(k^3))).

%e a(24) = 3 because we have [8, 8, 8], [8, 8, 1, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

%t nmax = 95; CoefficientList[Series[Product[1 + x^(2 k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]

%t Table[Length[Select[IntegerPartitions[n],AllTrue[Surd[#,3],IntegerQ] && Min[ Tally[ #][[All,2]]]>1&]],{n,0,100}] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run. *) (* _Harvey P. Dale_, May 03 2021 *)

%Y Cf. A000578, A003108, A007690, A161090.

%K nonn

%O 0,17

%A _Ilya Gutkovskiy_, Nov 27 2020