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A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

%I A356064 #7 Jul 29 2022 09:51:20
%S A356064 13,26,29,37,39,43,47,52,58,61,65,71,73,74,78,79,86,87,89,91,94,101,
%T A356064 104,107,111,113,116,117,122,129,130,137,139,141,142,143,145,146,148,
%U A356064 149,151,156,158,163,167,169,172,173,174,178,181,182,183,185,188,193
%N A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.
%C A356064 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A356064 These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.
%e A356064 The terms together with their prime indices begin:
%e A356064    13: {6}
%e A356064    26: {1,6}
%e A356064    29: {10}
%e A356064    37: {12}
%e A356064    39: {2,6}
%e A356064    43: {14}
%e A356064    47: {15}
%e A356064    52: {1,1,6}
%e A356064    58: {1,10}
%e A356064    61: {18}
%e A356064    65: {3,6}
%e A356064    71: {20}
%e A356064    73: {21}
%e A356064    74: {1,12}
%e A356064    78: {1,2,6}
%e A356064    79: {22}
%e A356064    86: {1,14}
%e A356064    87: {2,10}
%t A356064 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356064 Select[Range[100],!And@@PrimePowerQ/@DeleteCases[primeMS[#],1]&]
%Y A356064 Heinz numbers of the partitions counted by A023893.
%Y A356064 Allowing prime index 1 gives A356066.
%Y A356064 A000688 counts factorizations into prime-powers, strict A050361.
%Y A356064 A001222 counts prime-power divisors.
%Y A356064 A023894 counts partitions into prime-powers, strict A054685.
%Y A356064 A034699 gives the maximal prime-power divisor.
%Y A356064 A246655 lists the prime-powers (A000961 includes 1), towers A164336.
%Y A356064 A355742 chooses a prime-power divisor of each prime index.
%Y A356064 A355743 = numbers whose prime indices are prime-powers, squarefree A356065.
%Y A356064 Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.
%K A356064 nonn
%O A356064 1,1
%A A356064 _Gus Wiseman_, Jul 25 2022