cp's OEIS Frontend

A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.

%I A360953 #7 Mar 09 2023 23:08:36
%S A360953 1,4,9,12,16,25,30,48,49,63,64,70,81,108,121,154,165,169,192,256,270,
%T A360953 273,286,289,325,361,442,529,561,567,595,625,646,675,729,741,750,768,
%U A360953 841,874,931,961,972,1024,1045,1173,1334,1369,1495,1575,1653,1681,1750
%N A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.
%C A360953 Also numbers whose left half of prime indices (inclusive) adds up to half the total sum of prime indices.
%C A360953 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.
%e A360953 The terms together with their prime indices begin:
%e A360953      1: {}
%e A360953      4: {1,1}
%e A360953      9: {2,2}
%e A360953     12: {1,1,2}
%e A360953     16: {1,1,1,1}
%e A360953     25: {3,3}
%e A360953     30: {1,2,3}
%e A360953     48: {1,1,1,1,2}
%e A360953     49: {4,4}
%e A360953     63: {2,2,4}
%e A360953     64: {1,1,1,1,1,1}
%e A360953     70: {1,3,4}
%e A360953     81: {2,2,2,2}
%e A360953    108: {1,1,2,2,2}
%e A360953 For example, the prime indices of 1575 are {2,2,3,3,4}, with right half (exclusive) {3,4}, with sum 7, and the total sum of prime indices is 14, so 1575 is in the sequence.
%t A360953 Select[Range[100],With[{w=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[Take[w,-Floor[Length[w]/2]]]==Total[w]/2]&]
%Y A360953 The left version is A056798.
%Y A360953 The inclusive version is A056798.
%Y A360953 These partitions are counted by A360674.
%Y A360953 The left inclusive version is A360953 (this sequence).
%Y A360953 A112798 lists prime indices, length A001222, sum A056239, median* A360005.
%Y A360953 First for prime indices, second for partitions, third for prime factors:
%Y A360953 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360953 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360953 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360953 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360953 Cf. A000005, A000040, A001248, A026424, A359912, A360006, A360616, A360617, A360671, A360673.
%K A360953 nonn
%O A360953 1,2
%A A360953 _Gus Wiseman_, Mar 09 2023