A143348 a(n) = -(-1)^n times sum of divisors of n.

1, -3, 4, -7, 6, -12, 8, -15, 13, -18, 12, -28, 14, -24, 24, -31, 18, -39, 20, -42, 32, -36, 24, -60, 31, -42, 40, -56, 30, -72, 32, -63, 48, -54, 48, -91, 38, -60, 56, -90, 42, -96, 44, -84, 78, -72, 48, -124, 57, -93, 72, -98, 54, -120, 72, -120, 80, -90, 60, -168, 62, -96, 104, -127, 84, -144, 68, -126, 96

Offset: 1

Author

Michael Somos, Aug 09 2008

Keywords

• sign
• easy
• mult

Examples

			q - 3*q^2 + 4*q^3 - 7*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 15*q^8 + 13*q^9 + ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..10000
• László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.

Crossrefs

-(-1)^n * A000203(n) = a(n). A143337(n) = 24 * a(n) unless n=0.

Cf. A245058 (-Pi^2/48).

Programs

Mathematica

Table[-(-1)^n*DivisorSigma[1, n], {n, 69}] (* Michael De Vlieger, Aug 19 2017 *)

PARI

{a(n) = if( n<1, 0, -(-1)^n * sigma(n))}

Formula

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

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

Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 6 / 2^s + 4 / 4^s).

Sum_{k=1..n} a(k) ~ -(Pi^2/48) * n^2. - Amiram Eldar, Oct 28 2023