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

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Author

Gus Wiseman, Jul 25 2022

Keywords

• nonn

Examples

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

Crossrefs

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

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

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

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

These are the odd products of distinct elements of A302493.

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

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

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

This is the squarefree case of A355743, complement A356066.

A001222 counts prime-power divisors.

A005117 lists the squarefree numbers.

A034699 gives maximal prime-power divisor.

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

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Formula

Intersection of A005117 and A355743.