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Sum of odd prime factors of n, counted with multiplicity.
3

%I #16 Nov 03 2023 08:03:21

%S 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,

%T 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,

%U 16,7,22,29,59,8,61,31,13,0,18,14,67,17,26,12,71,6

%N Sum of odd prime factors of n, counted with multiplicity.

%C Contains all positive integers except 1, 2, 4.

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

%F a(2n) = a(n).

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

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

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

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

%o (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

%Y The compound version is A001414, triangle A331416.

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

%Y Odd-indexed terms are A100005.

%Y Positions of odd terms are A335657, even A036349.

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

%Y The even version is A366839 = 2*A001511.

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

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

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

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

%Y A162641 counts even prime exponents, odd A162642.

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

%Y A257992 counts even prime indices, odd A257991.

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

%K nonn,easy

%O 1,3

%A _Gus Wiseman_, Oct 27 2023