A161841 Number of factors, with repetition, in all distinct pairs (a <= b) such that a*b = n.

2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 6, 2, 6, 4, 4, 2, 8, 4, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 10, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 4, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 8, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 6, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 10

Offset: 1

Author

Omar E. Pol, Jun 23 2009

Keywords

• easy
• nonn

Examples

			a(16)=6 because there are three distinct pairs (a <= b) such that a*b = n: the pairs (1,16), (2,8) and (4,4). So the number of factors, with repetition, in all the pairs is equal to 6.
		

Links

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

Crossrefs

Cf. A000005, A010052, A038548, A161842.

Programs

Maple

seq(numtheory:-tau(n) + `if`(issqr(n),1,0), n = 1 .. 200); # Robert Israel, Dec 23 2015

Mathematica

2*Ceiling[DivisorSigma[0, Range[100]]/2] (* Paolo Xausa, Feb 05 2025 *)

Formula

a(n) = A000005(n) + A010052(n) = A038548(n)*2.

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1)*n + sqrt(n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2021