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A355743 Numbers whose prime indices are all prime-powers.

%I A355743 #13 Apr 02 2025 09:51:14
%S A355743 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 A355743 63,67,69,75,77,81,83,85,93,95,97,99,103,105,109,115,119,121,123,125,
%U A355743 127,131,133,135,147,153,155,157,159,161,165,171,175,177,179,187
%N A355743 Numbers whose prime indices are all prime-powers.
%C A355743 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 A355743 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}}.
%H A355743 Robert Price, <a href="/A355743/b355743.txt">Table of n, a(n) for n = 1..1410</a>
%e A355743 The terms together with their prime indices begin:
%e A355743    1: {}
%e A355743    3: {2}
%e A355743    5: {3}
%e A355743    7: {4}
%e A355743    9: {2,2}
%e A355743   11: {5}
%e A355743   15: {2,3}
%e A355743   17: {7}
%e A355743   19: {8}
%e A355743   21: {2,4}
%e A355743   23: {9}
%e A355743   25: {3,3}
%e A355743   27: {2,2,2}
%e A355743   31: {11}
%e A355743   33: {2,5}
%e A355743   35: {3,4}
%e A355743   41: {13}
%e A355743   45: {2,2,3}
%t A355743 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355743 Select[Range[100],And@@PrimePowerQ/@primeMS[#]&]
%Y A355743 The multiplicative version is A000688, strict A050361, coprime A354911.
%Y A355743 The case of only primes (not all prime-powers) is A076610, strict A302590.
%Y A355743 Allowing prime index 1 gives A302492.
%Y A355743 These are the products of elements of A302493.
%Y A355743 Requiring n to be a prime-power gives A302601.
%Y A355743 These are the positions of 1's in A355741.
%Y A355743 The squarefree case is A356065.
%Y A355743 The complement is A356066.
%Y A355743 A001222 counts prime-power divisors.
%Y A355743 A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.
%Y A355743 A034699 gives maximal prime-power divisor.
%Y A355743 A246655 lists the prime-powers (A000961 includes 1), towers A164336.
%Y A355743 A355742 chooses a prime-power divisor of each prime index.
%Y A355743 Cf. A085970, A106244, A279784, A295935, A355731, A356064.
%K A355743 nonn
%O A355743 1,2
%A A355743 _Gus Wiseman_, Jul 24 2022