A356065 - OEIS

%I #6 Jul 29 2022 09:51:24

%S 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 93,95,97,103,105,109,115,119,123,127,131,133,155,157,159,161,165,177,

%U 179,187,191,201,205,209,211,217,227,231,241,249,253,255,265,277

%N Squarefree numbers whose prime indices are all prime-powers.

%F Intersection of A005117 and A355743.

%e 105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

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

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

%Y The multiplicative version (factorizations) is A050361, non-strict A000688.

%Y Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.

%Y Counting twice-partitions of this type gives A279786, non-strict A279784.

%Y Counting twice-factorizations gives A295935, non-strict A296131.

%Y These are the odd products of distinct elements of A302493.

%Y Allowing prime index 1 gives A302496, non-strict A302492.

%Y The case of primes (instead of prime-powers) is A302590, non-strict A076610.

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

%Y This is the squarefree case of A355743, complement A356066.

%Y A001222 counts prime-power divisors.

%Y A005117 lists the squarefree numbers.

%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. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 25 2022