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%I #8 Jul 17 2021 07:00:50
%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,33,
%T 34,35,37,39,41,43,47,49,51,53,55,59,61,67,71,79
%N Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.
%e 48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).
%t nn=100;
%t ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
%t Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1
%Y Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.
%K nonn,fini,full
%O 1,2
%A _Gus Wiseman_, Dec 14 2018