A080736 - OEIS

A080736

Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.

1, 0, 2, 2, 4, 0, 6, 4, 6, 0, 10, 4, 12, 0, 8, 8, 16, 0, 18, 8, 12, 0, 22, 8, 20, 0, 18, 12, 28, 0, 30, 16, 20, 0, 24, 12, 36, 0, 24, 16, 40, 0, 42, 20, 24, 0, 46, 16, 42, 0, 32, 24, 52, 0, 40, 24, 36, 0, 58, 16, 60, 0, 36, 32, 48, 0, 66, 32, 44, 0, 70, 24, 72, 0, 40, 36, 60, 0, 78, 32

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OFFSET

1,3

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(A016825(n)) = 0; a(A000040(n)) = A000040(n) - 1. - Reinhard Zumkeller, Jun 11 2012

a(n) = if n mod 4 = 2 then 0 else A000010(n). - Reinhard Zumkeller, Jun 13 2012

From Amiram Eldar, Nov 02 2023: (Start)

Multiplicative with a(2) = 0, a(2^e) = 2^(e-1) for e >= 2, and a(p^e) = (p-1)*p^(e-1) for an odd prime p.

Dirichlet g.f.: (1 - 2^(1-s) + 1/(2^s-1)) * zeta(s-1) / zeta(s).

Sum_{k=1..n} a(k) ~ (5/(2*Pi^2)) * n^2. (End)