A334874 a(n) = sigma(1) - tau(2) + sigma(3) - tau(4) + sigma(5) - tau(6) + ... - (up to n).

1, -1, 3, 0, 6, 2, 10, 6, 19, 15, 27, 21, 35, 31, 55, 50, 68, 62, 82, 76, 108, 104, 128, 120, 151, 147, 187, 181, 211, 203, 235, 229, 277, 273, 321, 312, 350, 346, 402, 394, 436, 428, 472, 466, 544, 540, 588, 578, 635, 629, 701, 695, 749, 741, 813, 805, 885, 881, 941, 929

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_[k mod 2](k), where sigma_[0](n) = tau(n), the number of divisors of n and sigma_[1](n) = sigma(n), the sum of the divisors of n.

a(p^k) - a(p^k-1) = (p^(k+1)-1)/(p-1), where p is an odd prime and k is a positive integer. - Wesley Ivan Hurt, May 15 2020

Example

a(1) = sigma(1) = 1;
a(2) = sigma(1) - tau(2) = 1 - 2 = -1;
a(3) = sigma(1) - tau(2) + sigma(3) = 1 - 2 + 4 = 3;
a(4) = sigma(1) - tau(2) + sigma(3) - tau(4) = 1 - 2 + 4 - 3 = 0;

Maple

f:= proc(n) if n::odd then numtheory:-sigma(n) else -numtheory:-tau(n) fi end proc:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, May 15 2020

Mathematica

Table[Sum[(-1)^(k + 1)*DivisorSigma[Mod[k, 2], k], {k, n}], {n, 100}]

Prog

(PARI) a(n) = sum(k=1, n, (-1)^(k+1)*sigma(k, k % 2)); \\ Michel Marcus, May 14 2020

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A245466.

Sequence in context: A071126 A077187 A011079 * A339416 A226535 A005928

Adjacent sequences: A334871 A334872 A334873 * A334875 A334876 A334877

Keyword

sign

Author

Wesley Ivan Hurt, May 13 2020

Status

approved