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A344908
Sum of the distinct odd-indexed prime divisors, p_{2k-1}, of n.
3
0, 2, 0, 2, 5, 2, 0, 2, 0, 7, 11, 2, 0, 2, 5, 2, 17, 2, 0, 7, 0, 13, 23, 2, 5, 2, 0, 2, 0, 7, 31, 2, 11, 19, 5, 2, 0, 2, 0, 7, 41, 2, 0, 13, 5, 25, 47, 2, 0, 7, 17, 2, 0, 2, 16, 2, 0, 2, 59, 7, 0, 33, 0, 2, 5, 13, 67, 19, 23, 7, 0, 2, 73, 2, 5, 2, 11, 2, 0, 7, 0, 43, 83, 2, 22
OFFSET
1,2
COMMENTS
a(m) = 0 for m in A066207. - Michel Marcus, Jun 12 2021
Inverse Möbius transform of n * c(n) * (pi(n) mod 2), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 23 2024
LINKS
FORMULA
a(n) = Sum_{p|n} p * (pi(p) mod 2).
G.f.: Sum_{k>=1} prime(2*k-1) * x^prime(2*k-1) / (1 - x^prime(2*k-1)). - Ilya Gutkovskiy, Oct 24 2023
a(n) = Sum_{d|n} d * c(d) * (pi(d) mod 2), where c = A010051. - Wesley Ivan Hurt, Jun 23 2024
EXAMPLE
a(6) = Sum_{p|6} p * (pi(p) mod 2) = 2*(pi(2) mod 2) + 3*(pi(3) mod 2) = 2*1 + 3*0 = 2.
MATHEMATICA
Table[Sum[k*Mod[PrimePi[k], 2] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
PROG
(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (primepi(f[k, 1]) % 2, f[k, 1])); \\ Michel Marcus, Jun 12 2021
CROSSREFS
Cf. A344931 (sum of distinct even-indexed prime divisors).
Sequence in context: A358405 A209701 A158855 * A005074 A161564 A078182
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 02 2021
STATUS
approved