A352143 Numbers whose prime indices and conjugate prime indices are all odd.

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Author

Gus Wiseman, Mar 18 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, sum A056239, length A001222.

A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.

These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}
		

Crossrefs

The restriction to primes is A031368.

These partitions appear to be counted by A053253.

The even version is A066207^2.

For even instead of odd conjugate parts we get A066208^2.

The first condition alone (all odd indices) is A066208, counted by A000009.

The second condition alone is A346635, counted by A000009.

A055922 counts partitions with odd multiplicities, ranked by A268335.

A066207 = indices all even, counted by A035363 (complement A086543).

A109297 = same indices as exponents, counted by A114640.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

A162642 counts odd prime exponents, even A162641.

A238745 gives the Heinz number of the conjugate prime signature.

A257991 counts odd indices, even A257992.

A258116 ranks strict partitions with all odd parts, even A258117.

A351979 = odd indices and even multiplicities, counted by A035457.

A352140 = even indices and odd multiplicities, counted by A055922 aerated.

A352141 = even indices and even multiplicities, counted by A035444.

A352142 = odd indices and odd multiplicities, counted by A117958.

Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Formula

Intersection of A066208 and A346635.