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A268631 Number of ordered pairs (a,b) of positive integers less than n with the property that n divides ab. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Matthew McMullen, Feb 09 2016
STATUS
approved

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Last modified May 21 22:16 EDT 2024. Contains 372741 sequences. (Running on oeis4.)