A268631 Number of ordered pairs (a,b) of positive integers less than n with the property that n divides ab.

0, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 17, 0, 12, 16, 17, 0, 28, 0, 33, 24, 20, 0, 53, 16, 24, 28, 49, 0, 76, 0, 49, 40, 32, 48, 97, 0, 36, 48, 101, 0, 112, 0, 81, 100, 44, 0, 145, 36, 96, 64, 97, 0, 136, 80, 149, 72, 56, 0, 241, 0, 60, 148, 129, 96, 184, 0, 129, 88, 212

Offset

1,6

Comments

a(n)=0 iff n is prime or 1. a(n) is odd iff n is a multiple of 4.

Links

Table of n, a(n) for n=1..70.

Formula

a(n) = Sum_{k=1..n-1} (number of divisors of nk that are between k and n, exclusive).

a(n) = Sum_{k=1..n-1} (number of divisors of nk - 2*(number of divisors of nk that are <= k)).

a(n) = A006579(n) - (n-1). - Michel Marcus, Feb 09 2016

a(p^k) = (p(k-1)-k)*p^(k-1)+1 for prime p. - Chai Wah Wu, May 15 2022

Example

For n=10 the a(10)=8 ordered pairs are (2,5), (5,2), (4,5), (5,4), (5,6), (6,5), (5,8), and (8,5).

Mathematica

a[n_] := Sum[Sum[1, {i, Divisors[n*k]}] - 2*Sum[1, {i, TakeWhile[Divisors[n*k], # <= k &]}], {k, 1, n - 1}]

Prog

(PARI) a(n) = sum(k=1, n-1, sumdiv(n*k, d, (d > k) && (d < n))); \\ Michel Marcus, Feb 09 2016
(Python)
from math import prod
from sympy import factorint
def A268631(n): return 1 - 2*n + prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, May 15 2022

Crossrefs

Cf. A006579.

Sequence in context: A049247 A175621 A016578 * A335775 A308108 A320374

Adjacent sequences: A268628 A268629 A268630 * A268632 A268633 A268634

Keyword

nonn

Author

Matthew McMullen, Feb 09 2016

Status

approved