A143520 a(n) is n times number of divisors of n if n is odd, zero if n is twice odd, n times number of divisors of n/4 if n is divisible by 4.

1, 0, 6, 4, 10, 0, 14, 16, 27, 0, 22, 24, 26, 0, 60, 48, 34, 0, 38, 40, 84, 0, 46, 96, 75, 0, 108, 56, 58, 0, 62, 128, 132, 0, 140, 108, 74, 0, 156, 160, 82, 0, 86, 88, 270, 0, 94, 288, 147, 0, 204, 104, 106, 0, 220, 224, 228, 0, 118, 240, 122, 0, 378, 320, 260, 0, 134, 136

Offset: 1

Author

Michael Somos, Aug 22 2008

Keywords

• nonn
• mult

Examples

			q + 6*q^3 + 4*q^4 + 10*q^5 + 14*q^7 + 16*q^8 + 27*q^9 + 22*q^11 + 24*q^12 + ...
		

Links

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

Crossrefs

Cf. A001620, A027748, A038040, A112329, A124010.

Programs

Haskell

a143520 n = product $ zipWith (\p e -> (e + 2 * mod p 2 - 1) * p ^ e)
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jan 21 2014

Mathematica

Abs@Total[# (-1)^Divisors[#]] & /@ Range[68] (* George Beck, Oct 25 2014 *)
f[p_, e_] := (e + 1)*p^e; f[2, e_] := (e - 1)*2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 29 2022 *)

PARI

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; (e - (-1)^p) * p^e)))}

PARI

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, k * x^k / (1 - (-x)^k)^2, x*O(x^n)), n))}

Formula

a(n) is multiplicative with a(2^e) = (e-1) * 2^e if e>0, a(p^e) = (e+1) * p^e if p>2.

a(4*n + 2) = 0.

G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k)^2.

A038040(2*n + 1) = a(2*n + 1); 4 * A038040(n) = a(4*n).

From Amiram Eldar, Nov 29 2022: (Start)

a(n) = n * A112329(n).

Dirichlet g.f.: zeta(s-1)^2*(1+2^(3-2*s)-2^(2-s)).

Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma-1)*n^2/8, where gamma is Euler's constant (A001620). (End)