A366839 Sum of even prime factors of 2n, counted with multiplicity.

2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 12, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 14, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2

Offset: 1

Author

Gus Wiseman, Oct 25 2023

Keywords

• nonn

Examples

			The prime factors of 2*60 are {2,2,2,3,5}, of which the even factors are {2,2,2}, so a(60) = 6.
		

Crossrefs

A compound version is A001414, triangle A331416.

Dividing by 2 gives A001511.

Positions of 2's are A005408.

For count instead of sum we have A007814, odd version A087436.

The partition triangle for this statistic is A116598 aerated.

For indices we have A366531, halved A366533, triangle A113686/A174713.

The odd version is A366840.

A019507 lists numbers with (even factor sum) = (odd factor sum).

A066207 lists numbers with all even prime indices, counted by A035363.

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

A162641 counts even prime exponents, odd A162642.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

A257992 counts even prime indices, odd A257991.

A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).

Cf. A000720, A066208, A258117, A325698.

Programs

Mathematica

Table[2*Length[NestWhileList[#/2&,n,EvenQ]],{n,100}]

PARI

a(n) = 2 * valuation(n, 2) + 2; \\ Amiram Eldar, Sep 13 2024

Formula

a(n) = 2*A001511(n).

a(n) = A100006(n) - A366840(2n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Sep 13 2024