A379317 Positive integers with a unique even prime index.

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142

Offset: 1

Author

Gus Wiseman, Dec 29 2024

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.

Examples

			The terms together with their prime indices begin:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}
		

Crossrefs

Partitions of this type are counted by A038348 (strict A096911).

For all even parts we have A066207, counted by A035363 (strict A000700).

For no even parts we have A066208, counted by A000009 (strict A035457).

Positions of 1 in A257992.

A000040 lists the primes, differences A001223.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

Other counts of prime indices:

- A257991 odd, see A000041, A000070, A349158.

- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.

- A330944 nonprime, see A000586, A000607, A076610, A330945.

- A379300 composite, see A023895, A034891, A036497, A302540, A379301.

- A379311 old prime, see A204389, A320629, A379312-A379315.

Cf. A000720, A038550, A087436.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],EvenQ]]==1&]