A355743 Numbers whose prime indices are all prime-powers.

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 41, 45, 49, 51, 53, 55, 57, 59, 63, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 105, 109, 115, 119, 121, 123, 125, 127, 131, 133, 135, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 179, 187

Offset: 1

Author

Gus Wiseman, Jul 24 2022

Keywords

• nonn

Comments

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.

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}}.

Examples

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  35: {3,4}
  41: {13}
  45: {2,2,3}
		

Links

• Robert Price, Table of n, a(n) for n = 1..1410

Crossrefs

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

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

Allowing prime index 1 gives A302492.

These are the products of elements of A302493.

Requiring n to be a prime-power gives A302601.

These are the positions of 1's in A355741.

The squarefree case is A356065.

The complement is A356066.

A001222 counts prime-power divisors.

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

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. A085970, A106244, A279784, A295935, A355731, A356064.

Programs

Mathematica

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