A068074 a(n) = Sum_{d|n} (-1)^d*2^omega(n/d) where omega(x) is the number of distinct prime factors in the factorization of x.

-1, -1, -3, 1, -3, -3, -3, 3, -5, -3, -3, 3, -3, -3, -9, 5, -3, -5, -3, 3, -9, -3, -3, 9, -5, -3, -7, 3, -3, -9, -3, 7, -9, -3, -9, 5, -3, -3, -9, 9, -3, -9, -3, 3, -15, -3, -3, 15, -5, -5, -9, 3, -3, -7, -9, 9, -9, -3, -3, 9, -3, -3, -15, 9, -9, -9, -3, 3, -9, -9, -3, 15, -3, -3, -15, 3, -9, -9, -3, 15, -9, -3, -3, 9, -9, -3, -9, 9, -3

Offset

1,3

References

G. Tenenbaum and Jie Wu, Cours specialies No. 2: "Exercices corrigés de théorie analytique et probabiliste des nombres", Collection SMF, chap. II.7.1, p. 105.

Links

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

Formula

Asymptotic formula: Sum_{k=1..n} a(k)/k = -C*log(n)^2 with C = 3*log(2)/Pi^2.

a(n) = -tau(n^2) for odd n and 2*tau(n^2/4) - tau(n^2) for even n. b(n) = abs(a(n)) is multiplicative with b(2^e) = abs(2*e-3) and b(p^e) = 2*e+1 for an odd prime p. - Vladeta Jovovic, Apr 25 2002

a(n) = if n odd then -A048691(n) else 2*A048691(n/2) - A048691(n). - Reinhard Zumkeller, Jul 12 2012

Mathematica

a[n_?OddQ] := -DivisorSigma[0, n^2]; a[n_?EvenQ] := 2*DivisorSigma[0, n^2/4] - DivisorSigma[0, n^2]; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Nov 15 2011, after Vladeta Jovovic *)

Prog

(Haskell)
a068074 n | odd n     = - a048691 n
          | otherwise = 2 * a048691 (n `div` 2) - a048691 n
-- Reinhard Zumkeller, Jul 12 2012
(PARI) a(n) = sumdiv(n, d, (-1)^d*2^omega(n/d)); \\ Michel Marcus, Oct 08 2017

Crossrefs

Cf. A048691.

Sequence in context: A210146 A354198 A233654 * A063195 A334070 A025796

Adjacent sequences: A068071 A068072 A068073 * A068075 A068076 A068077

Keyword

easy,nice,sign

Author

Benoit Cloitre, Apr 14 2002

Status

approved