A379301 Positive integers whose prime indices include a unique composite number.

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

Offset: 1

Author

Gus Wiseman, Dec 25 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 prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.
		

Crossrefs

For no composite parts we have A302540, counted by A034891 (strict A036497).

For all composite parts we have A320629, counted by A023895 (strict A204389).

For a unique prime part we have A331915, counted by A379304 (strict A379305).

Positions of one in A379300.

Partitions of this type are counted by A379302 (strict A379303).

A000040 lists the prime numbers, differences A001223.

A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.

A055396 gives least prime index, greatest A061395.

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

A066247 is the characteristic function for the composite numbers.

A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.

Other counts of prime indices:

- A087436 postpositive, see A038550.

- A257991 odd, see A000041, A000070, A066207, A349158.

- A257992 even, see A000009, A038348, A066208, A379317.

- A257994 prime, see A000586, A000607, A076610, A331386.

- A330944 nonprime, see A002095, A096258, A320628, A330945.

- A379306 squarefree, see A302478, A379308, A379309, A379316.

- A379310 nonsquarefree, see A114374, A256012, A379307.

- A379311 old prime, see A379312-A379315.

Cf. A113646, A376680.

Programs

Mathematica

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