%I #8 Jul 28 2022 21:15:28

%S 1,3,5,7,9,11,15,17,19,21,23,25,27,31,33,35,41,45,49,51,53,55,57,59,

%T 63,67,69,75,77,81,83,85,93,95,97,99,103,105,109,115,119,121,123,125,

%U 127,131,133,135,147,153,155,157,159,161,165,171,175,177,179,187

%N Numbers whose prime indices are all prime-powers.

%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.

%C Also MM-numbers of multiset partitions into constant multisets, where the multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

%e The terms together with their prime indices begin:

%e 1: {}

%e 3: {2}

%e 5: {3}

%e 7: {4}

%e 9: {2,2}

%e 11: {5}

%e 15: {2,3}

%e 17: {7}

%e 19: {8}

%e 21: {2,4}

%e 23: {9}

%e 25: {3,3}

%e 27: {2,2,2}

%e 31: {11}

%e 33: {2,5}

%e 35: {3,4}

%e 41: {13}

%e 45: {2,2,3}

%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 multiplicative version is A000688, strict A050361, coprime A354911.

%Y The case of only primes (not all prime-powers) is A076610, strict A302590.

%Y Allowing prime index 1 gives A302492.

%Y These are the products of elements of A302493.

%Y Requiring n to be a prime-power gives A302601.

%Y These are the positions of 1's in A355741.

%Y The squarefree case is A356065.

%Y The complement is A356066.

%Y A001222 counts prime-power divisors.

%Y A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.

%Y A034699 gives 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. A085970, A106244, A279784, A295935, A355731, A356064.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 24 2022