A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

1, 12, 70, 90, 112, 144, 286, 325, 462, 520, 525, 594, 646, 675, 832, 840, 1045, 1080, 1326, 1334, 1344, 1666, 1672, 1728, 1900, 2142, 2145, 2294, 2465, 2622, 2695, 2754, 3040, 3432, 3465, 3509, 3526, 3900, 3944, 4186, 4255, 4312, 4455, 4845, 4864, 4900, 4982

Offset: 1

Author

Gus Wiseman, Oct 23 2023

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: {}
    12: {1,1,2}
    70: {1,3,4}
    90: {1,2,2,3}
   112: {1,1,1,1,4}
   144: {1,1,1,1,2,2}
   286: {1,5,6}
   325: {3,3,6}
   462: {1,2,4,5}
   520: {1,1,1,3,6}
   525: {2,3,3,4}
   594: {1,2,2,2,5}
   646: {1,7,8}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
   840: {1,1,1,2,3,4}
For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.
		

Crossrefs

For prime factors instead of indices we have A019507.

Partitions of this type are counted by A239261.

For count instead of sum we have A325698, distinct A325700.

The LHS (sum of odd prime indices) is A366528, triangle A113685.

The RHS (sum of even prime indices) is A366531, triangle A113686.

These are the positions of zeros in A366749.

A000009 counts partitions into odd parts, ranked by A066208.

A035363 counts partitions into even parts, ranked by A066207.

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

A257991 counts odd prime indices, even A257992.

A300061 lists numbers with even sum of prime indices, odd A300063.

Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.

Programs

Mathematica

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

Formula

These are numbers k such that A346697(k) = A346698(k).