A241534 Number of integer arithmetic means of 2 distinct divisors of n.

0, 0, 1, 1, 1, 2, 1, 3, 3, 2, 1, 7, 1, 2, 6, 6, 1, 6, 1, 7, 6, 2, 1, 16, 3, 2, 6, 7, 1, 12, 1, 10, 6, 2, 6, 18, 1, 2, 6, 16, 1, 12, 1, 7, 15, 2, 1, 29, 3, 6, 6, 7, 1, 12, 6, 16, 6, 2, 1, 34, 1, 2, 15, 15, 6, 12, 1, 7, 6, 12, 1, 39, 1, 2, 15, 7, 6, 12, 1, 29

Offset: 1

Author

Ilya Lopatin and Juri-Stepan Gerasimov, May 08 2014

Keywords

• nonn

Examples

			Triangle T(n, k) starts for n > 2:
2,
3,
3,
2, 4,
4,
3, 5, 6,
2, 5, 6;
where T(n, k) = the values of k such that 2k = q + g; q, g are distinct divisors of n.
a(20) = 7 because (1,5), (2,4), (2,10), (2,20), (4,10), (4,20) and (10,20) are the 7 values of (g,q) such that (g+q)/2 is an integer. - _Colin Barker_, May 10 2014
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A027750.

Programs

Mathematica

Table[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Oct 06 2020 *)

PARI

a(n) = c=0; fordiv(n, g, fordiv(n, q, if(gColin Barker, May 10 2014

Formula

a(n) = Sum_{d1|n, d2|n, d1 < d2} (1 - ceiling((d1+d2)/2) + floor((d1+d2)/2)). - Wesley Ivan Hurt, Oct 06 2020

Extensions

Several incorrect terms corrected, and more terms added by Colin Barker, May 10 2014