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%I #20 Apr 28 2023 08:20:40
%S 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,
%T 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,
%U 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
%N Number of distinct solutions to the congruence x(1)*x(2) == 0 (mod n), with x() only in 1..n-1.
%C 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
%H R. H. Hardin, <a href="/A180772/b180772.txt">Table of n, a(n) for n=1..10000</a>
%F a(n) = A174088(n) - n = ( A018804(n) + A000188(n) )/2 - n.
%e 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.
%t 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 *)
%Y Column 1 of A180782.
%Y Cf. A000188, A018804, A174088.
%K nonn
%O 1,6
%A _R. H. Hardin_, formula from _Max Alekseyev_ in the Sequence Fans Mailing List, Sep 20 2010
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