A230476 a(n) = Sum_{i=1..n} d(8*i+1) - Sum_{i=1..n} d(2*i+1), where d(n) = A000005(n) is the number of divisors of n.

1, 1, 2, 3, 3, 4, 4, 6, 6, 7, 7, 6, 10, 10, 11, 11, 9, 11, 13, 15, 16, 14, 16, 15, 15, 17, 17, 22, 22, 22, 20, 18, 20, 24, 24, 25, 27, 27, 27, 26, 28, 26, 30, 30, 29, 31, 31, 37, 35, 35, 35, 31, 35, 35, 40, 40, 38, 40, 40, 41, 41, 41, 43, 47, 47, 46, 42, 44, 46, 50, 48, 46, 52, 52, 52, 54, 52, 55, 55, 53, 55, 53, 59, 58, 56, 58

Offset

1,3

Comments

Cimadevilla proved that a(n) >= 0. That is surprising because d(8*i+1) - d(2*i+1) < 0 for i = 12, 17, 22, 24, 31, 32, 40, 42, 45, 49, 52, 57, 66, 67, 71, 72, 77, 80, 82, 84, 85, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. See inequalities (1.5).

Formula

a(n) = Sum_{i=1..n} (d(8*i+1) - d(2*i+1)) = A230293(n) + A230294(n).

a(n) = log(2) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

Example

The divisors of 8*1 + 1 = 9 are 1, 3, 9 and those of 2*1 + 1 = 3 are 1, 3, so a(1) = d(9) - d(3) = 3 - 2 = 1.

Mathematica

Table[Sum[ DivisorSigma[0, 8 i + 1] - DivisorSigma[0, 2 i + 1], {i, n}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, n, numdiv(8*i+1) - numdiv(2*i+1)); \\ Michel Marcus, Jun 19 2015

Crossrefs

Cf. A000005, A002162, A230290, A230291, A230292, A230293, A230295, A230296.

Sequence in context: A344870 A318283 A105677 * A103297 A274017 A267597

Adjacent sequences: A230473 A230474 A230475 * A230477 A230478 A230479

Keyword

nonn

Author

Jonathan Sondow, Oct 20 2013

Status

approved