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

1,6

Links

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

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

Example

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

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 *)

Prog

(PARI) a(n) = c=0; fordiv(n, g, fordiv(n, q, if(g<q && (g+q)%2==0, c++))); c \\ Colin Barker, May 10 2014

Crossrefs

Cf. A027750.

Sequence in context: A047679 A179480 A245326 * A337137 A035050 A198790

Adjacent sequences: A241531 A241532 A241533 * A241535 A241536 A241537

Keyword

nonn

Author

Ilya Lopatin and Juri-Stepan Gerasimov, May 08 2014

Extensions

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

Status

approved