A356066 Numbers with a prime index that is not a prime-power. Complement of A355743.

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 52, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 87, 88, 89, 90, 91, 92, 94, 96, 98, 100, 101

Offset

1,1

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.

Links

Table of n, a(n) for n=1..65.

Formula

Union of A299174 and A356064.

Example

The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}

Mathematica

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

Crossrefs

The complement is A355743, counted by A023894.

The squarefree complement is A356065, counted by A054685.

Allowing prime index 1 gives A356064, complement A302492.

A000688 counts factorizations into prime-powers, strict A050361.

A001222 counts prime-power divisors.

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.

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

Sequence in context: A301454 A161602 A359496 * A285591 A327210 A274611

Adjacent sequences: A356063 A356064 A356065 * A356067 A356068 A356069

Keyword

nonn

Author

Gus Wiseman, Jul 31 2022

Status

approved