A356064 Numbers with a prime index other than 1 that is not a prime-power. Complement of A302492.

13, 26, 29, 37, 39, 43, 47, 52, 58, 61, 65, 71, 73, 74, 78, 79, 86, 87, 89, 91, 94, 101, 104, 107, 111, 113, 116, 117, 122, 129, 130, 137, 139, 141, 142, 143, 145, 146, 148, 149, 151, 156, 158, 163, 167, 169, 172, 173, 174, 178, 181, 182, 183, 185, 188, 193

Offset: 1

Author

Gus Wiseman, Jul 25 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.

These are numbers divisible by a prime number not of the form prime(q^k) where q is a prime number and k >= 1.

Examples

			The terms together with their prime indices begin:
   13: {6}
   26: {1,6}
   29: {10}
   37: {12}
   39: {2,6}
   43: {14}
   47: {15}
   52: {1,1,6}
   58: {1,10}
   61: {18}
   65: {3,6}
   71: {20}
   73: {21}
   74: {1,12}
   78: {1,2,6}
   79: {22}
   86: {1,14}
   87: {2,10}
		

Crossrefs

Heinz numbers of the partitions counted by A023893.

Allowing prime index 1 gives A356066.

A000688 counts factorizations into prime-powers, strict A050361.

A001222 counts prime-power divisors.

A023894 counts partitions into prime-powers, strict A054685.

A034699 gives the maximal prime-power divisor.

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

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

A355743 = numbers whose prime indices are prime-powers, squarefree A356065.

Cf. A076610, A085970, A106244, A302492, A302493, A302601, A330946, A354911.

Programs

Mathematica

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