cp's OEIS Frontend

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

%I A366748 #9 Oct 23 2023 21:43:07
%S A366748 1,12,70,90,112,144,286,325,462,520,525,594,646,675,832,840,1045,1080,
%T A366748 1326,1334,1344,1666,1672,1728,1900,2142,2145,2294,2465,2622,2695,
%U A366748 2754,3040,3432,3465,3509,3526,3900,3944,4186,4255,4312,4455,4845,4864,4900,4982
%N A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).
%C A366748 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 A366748 These are numbers k such that A346697(k) = A346698(k).
%e A366748 The terms together with their prime indices begin:
%e A366748      1: {}
%e A366748     12: {1,1,2}
%e A366748     70: {1,3,4}
%e A366748     90: {1,2,2,3}
%e A366748    112: {1,1,1,1,4}
%e A366748    144: {1,1,1,1,2,2}
%e A366748    286: {1,5,6}
%e A366748    325: {3,3,6}
%e A366748    462: {1,2,4,5}
%e A366748    520: {1,1,1,3,6}
%e A366748    525: {2,3,3,4}
%e A366748    594: {1,2,2,2,5}
%e A366748    646: {1,7,8}
%e A366748    675: {2,2,2,3,3}
%e A366748    832: {1,1,1,1,1,1,6}
%e A366748    840: {1,1,1,2,3,4}
%e A366748 For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.
%t A366748 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366748 Select[Range[1000], Total[Select[prix[#],OddQ]]==Total[Select[prix[#],EvenQ]]&]
%Y A366748 For prime factors instead of indices we have A019507.
%Y A366748 Partitions of this type are counted by A239261.
%Y A366748 For count instead of sum we have A325698, distinct A325700.
%Y A366748 The LHS (sum of odd prime indices) is A366528, triangle A113685.
%Y A366748 The RHS (sum of even prime indices) is A366531, triangle A113686.
%Y A366748 These are the positions of zeros in A366749.
%Y A366748 A000009 counts partitions into odd parts, ranked by A066208.
%Y A366748 A035363 counts partitions into even parts, ranked by A066207.
%Y A366748 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A366748 A257991 counts odd prime indices, even A257992.
%Y A366748 A300061 lists numbers with even sum of prime indices, odd A300063.
%Y A366748 Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.
%K A366748 nonn
%O A366748 1,2
%A A366748 _Gus Wiseman_, Oct 23 2023