cp's OEIS Frontend

A356065 Squarefree numbers whose prime indices are all prime-powers.

%I A356065 #6 Jul 29 2022 09:51:24
%S A356065 1,3,5,7,11,15,17,19,21,23,31,33,35,41,51,53,55,57,59,67,69,77,83,85,
%T A356065 93,95,97,103,105,109,115,119,123,127,131,133,155,157,159,161,165,177,
%U A356065 179,187,191,201,205,209,211,217,227,231,241,249,253,255,265,277
%N A356065 Squarefree numbers whose prime indices are all prime-powers.
%F A356065 Intersection of A005117 and A355743.
%e A356065 105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.
%t A356065 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356065 Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]
%Y A356065 The multiplicative version (factorizations) is A050361, non-strict A000688.
%Y A356065 Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
%Y A356065 Counting twice-partitions of this type gives A279786, non-strict A279784.
%Y A356065 Counting twice-factorizations gives A295935, non-strict A296131.
%Y A356065 These are the odd products of distinct elements of A302493.
%Y A356065 Allowing prime index 1 gives A302496, non-strict A302492.
%Y A356065 The case of primes (instead of prime-powers) is A302590, non-strict A076610.
%Y A356065 These are the squarefree positions of 1's in A355741.
%Y A356065 This is the squarefree case of A355743, complement A356066.
%Y A356065 A001222 counts prime-power divisors.
%Y A356065 A005117 lists the squarefree numbers.
%Y A356065 A034699 gives maximal prime-power divisor.
%Y A356065 A246655 lists the prime-powers (A000961 includes 1), towers A164336.
%Y A356065 A355742 chooses a prime-power divisor of each prime index.
%Y A356065 Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.
%K A356065 nonn
%O A356065 1,2
%A A356065 _Gus Wiseman_, Jul 25 2022