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

1, 4, 9, 12, 16, 25, 30, 48, 49, 63, 64, 70, 81, 108, 121, 154, 165, 169, 192, 256, 270, 273, 286, 289, 325, 361, 442, 529, 561, 567, 595, 625, 646, 675, 729, 741, 750, 768, 841, 874, 931, 961, 972, 1024, 1045, 1173, 1334, 1369, 1495, 1575, 1653, 1681, 1750

Offset: 1

Author

Gus Wiseman, Mar 09 2023

Keywords

• nonn

Comments

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

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: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    30: {1,2,3}
    48: {1,1,1,1,2}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    70: {1,3,4}
    81: {2,2,2,2}
   108: {1,1,2,2,2}
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.
		

Crossrefs

The left version is A056798.

The inclusive version is A056798.

These partitions are counted by A360674.

The left inclusive version is A360953 (this sequence).

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

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

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

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

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

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

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

Programs

Mathematica

Select[Range[100],With[{w=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[Take[w,-Floor[Length[w]/2]]]==Total[w]/2]&]