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

Author

Jonathan Sondow, Oct 20 2013

Keywords

• nonn

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, ...

Examples

			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.
		

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).

Crossrefs

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

Programs

Mathematica

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

PARI

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

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