A366748 - OEIS

%I #9 Oct 23 2023 21:43:07

%S 1,12,70,90,112,144,286,325,462,520,525,594,646,675,832,840,1045,1080,

%T 1326,1334,1344,1666,1672,1728,1900,2142,2145,2294,2465,2622,2695,

%U 2754,3040,3432,3465,3509,3526,3900,3944,4186,4255,4312,4455,4845,4864,4900,4982

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

%C 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.

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

%e The terms together with their prime indices begin:

%e      1: {}

%e     12: {1,1,2}

%e     70: {1,3,4}

%e     90: {1,2,2,3}

%e    112: {1,1,1,1,4}

%e    144: {1,1,1,1,2,2}

%e    286: {1,5,6}

%e    325: {3,3,6}

%e    462: {1,2,4,5}

%e    520: {1,1,1,3,6}

%e    525: {2,3,3,4}

%e    594: {1,2,2,2,5}

%e    646: {1,7,8}

%e    675: {2,2,2,3,3}

%e    832: {1,1,1,1,1,1,6}

%e    840: {1,1,1,2,3,4}

%e For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000], Total[Select[prix[#],OddQ]]==Total[Select[prix[#],EvenQ]]&]

%Y For prime factors instead of indices we have A019507.

%Y Partitions of this type are counted by A239261.

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

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

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

%Y These are the positions of zeros in A366749.

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A035363 counts partitions into even parts, ranked by A066207.

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

%Y A257991 counts odd prime indices, even A257992.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 23 2023