

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
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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..n1} (number of divisors of nk that are between k and n, exclusive).
a(n) = Sum_{k=1..n1} (number of divisors of nk  2*(number of divisors of nk that are <= k)).
a(p^k) = (p(k1)k)*p^(k1)+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, n1, 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**(e1)*((p1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, May 15 2022


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



