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Numbers with a prime index that is not a prime-power. Complement of A355743.
5

%I #6 Aug 01 2022 08:21:24

%S 2,4,6,8,10,12,13,14,16,18,20,22,24,26,28,29,30,32,34,36,37,38,39,40,

%T 42,43,44,46,47,48,50,52,54,56,58,60,61,62,64,65,66,68,70,71,72,73,74,

%U 76,78,79,80,82,84,86,87,88,89,90,91,92,94,96,98,100,101

%N Numbers with a prime index that is not a prime-power. Complement of A355743.

%C 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.

%F Union of A299174 and A356064.

%e The terms together with their prime indices begin:

%e 2: {1}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 10: {1,3}

%e 12: {1,1,2}

%e 13: {6}

%e 14: {1,4}

%e 16: {1,1,1,1}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 22: {1,5}

%e 24: {1,1,1,2}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],!And@@PrimePowerQ/@primeMS[#]&]

%Y The complement is A355743, counted by A023894.

%Y The squarefree complement is A356065, counted by A054685.

%Y Allowing prime index 1 gives A356064, complement A302492.

%Y A000688 counts factorizations into prime-powers, strict A050361.

%Y A001222 counts prime-power divisors.

%Y A034699 gives the maximal prime-power divisor.

%Y A246655 lists the prime-powers (A000961 includes 1), towers A164336.

%Y A355742 chooses a prime-power divisor of each prime index.

%Y Cf. A076610, A085970, A106244, A302493, A302601, A330946, A354911.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 31 2022