A037213 Expansion of Sum_{n>=0} n*q^(n^2).

0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• changed

Comments

Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. - Mitch Harris, Jun 09 2005

a(n) is the square root of n if n is square, zero otherwise. - Carl R. White, May 23 2009

Möbius transform of A069290(n). - Wesley Ivan Hurt, Jul 09 2025

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000196, A008683 (mu), A010052, A066839, A069290, A070039.

Programs

Haskell

a037213 n = if n == r ^ 2 then r else 0  where r = a000196 n
a037213_list = zipWith (*) a010052_list a000196_list
-- Reinhard Zumkeller, Nov 09 2012

Mathematica

Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 0, 100}] (* Geoffrey Critzer, Feb 21 2015 *)

PARI

a(n) = if (issquare(n), sqrtint(n), 0); \\ Michel Marcus, Aug 21 2025

Formula

Dirichlet generating function: zeta(2*s-1). - Franklin T. Adams-Watters, Sep 11 2005

a(n) = sqrt(n) * floor( cos^2(Pi * sqrt(n)) ). - Carl R. White, May 23 2009

a(n) = A000196(n) * A010052(n). - Reinhard Zumkeller, Jan 27 2010

Sum_{k=1..n} a(k) ~ n/2. - Vaclav Kotesovec, Aug 19 2021

a(n) = A066839(n) - A070039(n). - Ridouane Oudra, Jun 24 2025

a(n) = Sum_{d|n} A069290(d) * mu(n/d). - Wesley Ivan Hurt, Jul 09 2025