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Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.
5

%I #7 Mar 09 2023 23:08:36

%S 1,4,9,12,16,25,30,48,49,63,64,70,81,108,121,154,165,169,192,256,270,

%T 273,286,289,325,361,442,529,561,567,595,625,646,675,729,741,750,768,

%U 841,874,931,961,972,1024,1045,1173,1334,1369,1495,1575,1653,1681,1750

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

%C Also numbers whose left half of prime indices (inclusive) adds up to half the total sum of prime indices.

%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.

%e The terms together with their prime indices begin:

%e 1: {}

%e 4: {1,1}

%e 9: {2,2}

%e 12: {1,1,2}

%e 16: {1,1,1,1}

%e 25: {3,3}

%e 30: {1,2,3}

%e 48: {1,1,1,1,2}

%e 49: {4,4}

%e 63: {2,2,4}

%e 64: {1,1,1,1,1,1}

%e 70: {1,3,4}

%e 81: {2,2,2,2}

%e 108: {1,1,2,2,2}

%e 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 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 The left version is A056798.

%Y The inclusive version is A056798.

%Y These partitions are counted by A360674.

%Y The left inclusive version is A360953 (this sequence).

%Y A112798 lists prime indices, length A001222, sum A056239, median* A360005.

%Y First for prime indices, second for partitions, third for prime factors:

%Y - A360676 gives left sum (exclusive), counted by A360672, product A361200.

%Y - A360677 gives right sum (exclusive), counted by A360675, product A361201.

%Y - A360678 gives left sum (inclusive), counted by A360675, product A347043.

%Y - A360679 gives right sum (inclusive), counted by A360672, product A347044.

%Y Cf. A000005, A000040, A001248, A026424, A359912, A360006, A360616, A360617, A360671, A360673.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 09 2023