A366840 Sum of odd prime factors of n, counted with multiplicity.

0, 0, 3, 0, 5, 3, 7, 0, 6, 5, 11, 3, 13, 7, 8, 0, 17, 6, 19, 5, 10, 11, 23, 3, 10, 13, 9, 7, 29, 8, 31, 0, 14, 17, 12, 6, 37, 19, 16, 5, 41, 10, 43, 11, 11, 23, 47, 3, 14, 10, 20, 13, 53, 9, 16, 7, 22, 29, 59, 8, 61, 31, 13, 0, 18, 14, 67, 17, 26, 12, 71, 6

Offset: 1

Author

Gus Wiseman, Oct 27 2023

Keywords

• nonn
• easy

Comments

Contains all positive integers except 1, 2, 4.

Examples

			The prime factors of 60 are {2,2,2,3,5}, of which the odd factors are {3,5}, so a(60) = 8.
		

Crossrefs

The compound version is A001414, triangle A331416.

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

Odd-indexed terms are A100005.

Positions of odd terms are A335657, even A036349.

For prime indices we have A366528, triangle A113685 (without zeros A365067)

The even version is A366839 = 2*A001511.

The partition triangle for this statistic is A366851, even version A116598.

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.

Cf. A000009, A066208, A113686, A174713, A258117, A325698, A366531, A366850.

Programs

Mathematica

Table[Total[Times@@@DeleteCases[If[n==1,{}, FactorInteger[n]],{2,_}]],{n,100}]

PARI

a(n) = my(f=factor(n), j=if(n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ Michel Marcus, Oct 30 2023

Formula

a(n) = A100006(n) - A366839(n).

a(2n) = a(n).

a(2n-1) = A001414(2n-1) = A100005(n).

Completely additive with a(2^e) = 0 and a(p^e) = e*p for an odd prime p. - Amiram Eldar, Nov 03 2023