A180772 Number of distinct solutions to the congruence x(1)*x(2) == 0 (mod n), with x() only in 1..n-1.

0, 0, 0, 1, 0, 2, 0, 3, 3, 4, 0, 9, 0, 6, 8, 10, 0, 15, 0, 17, 12, 10, 0, 27, 10, 12, 15, 25, 0, 38, 0, 26, 20, 16, 24, 51, 0, 18, 24, 51, 0, 56, 0, 41, 51, 22, 0, 74, 21, 50, 32, 49, 0, 69, 40, 75, 36, 28, 0, 121, 0, 30, 75, 68, 48, 92, 0, 65, 44, 106, 0, 141, 0, 36, 90, 73, 60, 110, 0, 138

Offset

1,6

Comments

Also, number of ordered pairs (a, b) with 0 < a <= b such that a*b = c*n + d and c = d where 0 < a, b, c, d < n. - Naohiro Nomoto, Oct 02 2021

Links

R. H. Hardin, Table of n, a(n) for n=1..10000

Formula

a(n) = A174088(n) - n = ( A018804(n) + A000188(n) )/2 - n.

Example

The a(12)=9 solutions for product of a single 1..11 pair == 0 (mod 12) are 2*6, 3*4, 3*8, 4*6, 4*9, 6*6, 6*8, 6*10, and 8*9.

Mathematica

f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := p^Floor[e/2]; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct)/2 - n; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

Crossrefs

Column 1 of A180782.

Cf. A000188, A018804, A174088.

Sequence in context: A239313 A046667 A108407 * A370594 A291304 A245332

Adjacent sequences: A180769 A180770 A180771 * A180773 A180774 A180775

Keyword

nonn

Author

R. H. Hardin, formula from Max Alekseyev in the Sequence Fans Mailing List, Sep 20 2010

Status

approved