A357707 Numbers whose prime indices have equal number of parts congruent to each of 1 and 3 (mod 4).

1, 3, 7, 9, 10, 13, 19, 21, 27, 29, 30, 34, 37, 39, 43, 49, 53, 55, 57, 61, 62, 63, 70, 71, 79, 81, 87, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 117, 129, 130, 131, 133, 134, 139, 147, 151, 159, 163, 165, 166, 169, 171, 173, 181, 183, 186, 187, 189

Offset: 1

Author

Gus Wiseman, Oct 12 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.

Examples

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     7: {4}
     9: {2,2}
    10: {1,3}
    13: {6}
    19: {8}
    21: {2,4}
    27: {2,2,2}
    29: {10}
    30: {1,2,3}
		

Crossrefs

These partitions are counted by A035544.

Includes A066207 = products of primes of even index.

The conjugate partitions are ranked by A357636, reverse A357632.

The conjugate reverse version is A357640 (aerated).

A056239 adds up prime indices, row sums of A112798.

A316524 gives alternating sum of prime indices, reverse A344616.

A344651 counts partitions by alternating sum, ordered A097805.

A357705 counts reversed partitions by skew-alternating sum, half A357704.

Cf. A035363, A035550, A035594, A053251, A298311, A357486, A357623, A357638.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],?(Mod[#,4]==1&)]==Count[primeMS[#],?(Mod[#,4]==3&)]&]