A379312 Positive integers whose prime indices include a unique 1 or prime number.

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Author

Gus Wiseman, Dec 28 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:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}
		

Crossrefs

These "old" primes are listed by A008578.

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 ones in A379311, see A379313.

Partitions of this type are counted by A379314, strict A379315.

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.

A080339 is the characteristic function for the old prime numbers.

A376682 gives k-th differences of old prime numbers, see A030016, A075526.

Other counts of prime indices:

- 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.

Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]