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URL: https://oeis.org/A087726

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A087726
Number of elements X in the matrix ring M_2(Z_n) such that X^2 == 0 mod n.
2
1, 4, 9, 28, 25, 36, 49, 112, 153, 100, 121, 252, 169, 196, 225, 640, 289, 612, 361, 700, 441, 484, 529, 1008, 1225, 676, 1377, 1372, 841, 900, 961, 2560, 1089, 1156, 1225, 4284, 1369, 1444, 1521, 2800, 1681, 1764, 1849, 3388, 3825, 2116, 2209, 5760, 4753, 4900, 2601, 4732
OFFSET
1,2
COMMENTS
Conjecture: a(n)=n^2 if and only if n is squarefree. [Ben Branman, Mar 22 2013]
Preceding conjecture is true in the case where n is squarefree. - Eric M. Schmidt, Mar 23 2013
It appears that a(p^k) = (1+3*p^2 + 2*k*(p^2-1) + (-1)^k*(p^2-1))*p^(2*k-2)/4 for primes p. Since the sequence is multiplicative, this would imply the conjecture. - Robert Israel, Jun 10 2015
A proof of the formula for k=1 can be done easily (see pdf). - Manfred Scheucher, Jun 10 2015
LINKS
MAPLE
f:= proc(n)
local tot, S, a, mult, sa, d, ad, g, cands;
tot:= 0;
S:= ListTools:-Classify(t -> t^2 mod n, [$0..n-1]);
for a in numtheory:-divisors(n) do
mult:= numtheory:-phi(n/a);
sa:= a^2 mod n;
for d in S[sa] do
g:= igcd(a+d, n);
cands:= [seq(i*n/g, i=0..g-1)];
tot:= tot + mult * numboccur(sa, [seq(seq(s*t, s=cands), t=cands)] mod n);
od
od;
tot
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2015
MATHEMATICA
a[m_] := Count[Table[Mod[MatrixPower[Partition[IntegerDigits[n, m, 4], 2], 2], m] == {{0, 0}, {0, 0}}, {n, 0, m^4 - 1}], True]; Table[a[n], {n, 2, 30}] (* Ben Branman, Mar 22 2013 *)
PROG
(C)
#include<stdio.h>
#include<stdlib.h>
int main(int argc, char** argv)
{
long ct = 0;
int n = atoi(argv[1]);
int a, b, c, d;
for(a=0; a<n; a++)
{
for(b=0; b<n; b++)
{
for(c=0; c<n; c++)
{
if((a*a+b*c)%n != 0) continue;
for(d=0; d<n; d++)
{
if((b*c+d*d)%n != 0) continue;
if((a*b+b*d)%n != 0) continue;
if((c*a+d*c)%n != 0) continue;
ct++;
}
}
}
}
printf("%d %ld\n", n, ct);
return 0;
}
/* Manfred Scheucher, Jun 09 2015 */
CROSSREFS
Sequence in context: A272089 A270720 A272280 * A270891 A272313 A239668
KEYWORD
mult,nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 28 2003
EXTENSIONS
More terms from Ben Branman, Mar 22 2013
More terms from Manfred Scheucher, Jun 09 2015
STATUS
approved