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Numbers k such that every integer partition of k contains a 1 or a prime power.
2

%I #10 Jul 17 2021 07:00:15

%S 1,2,3,4,5,7,8,9,11,13,17,19,23

%N Numbers k such that every integer partition of k contains a 1 or a prime power.

%e 24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

%t nn=100;

%t ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];

%t Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

%Y Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

%K nonn,fini,full

%O 1,2

%A _Gus Wiseman_, Dec 14 2018