A316985 Number of solutions to x^2 + y^2 - z^2 == 1 (mod n).
1, 4, 12, 24, 30, 48, 56, 128, 108, 120, 132, 288, 182, 224, 360, 512, 306, 432, 380, 720, 672, 528, 552, 1536, 750, 728, 972, 1344, 870, 1440, 992, 2048, 1584, 1224, 1680, 2592, 1406, 1520, 2184, 3840, 1722, 2688, 1892, 3168, 3240, 2208, 2256, 6144, 2744, 3000
Offset: 1
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Programs
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Maple
g:= proc(t) if t[1]=2 then 2^(2*t[2]+1) else (t[1]+1)*t[1]^(2*t[2]-1) fi end proc: g([2,1]):= 4: g([2,2]):= 24: seq(convert(map(g, ifactors(n)[2]),`*`),n=1..100); # Robert Israel, Jul 20 2018
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Mathematica
a[n_] := a[n] = If[PrimeQ[n], n(n+1), Times @@ (Which[#[[1]] == 2 && #[[2]] == 1, 4, #[[1]] == 2 && #[[2]] == 2, 24, #[[1]] == 2, 2^(2 #[[2]]+1), True, (#[[1]]+1) #[[1]]^(2 #[[2]]-1)]& /@ FactorInteger[n])]; a[1] = 1; a[2] = 4; Array[a, 50] (* Jean-François Alcover, Jul 25 2018 and slightly modified by Robert G. Wilson v, Jul 25 2018 *)
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PARI
M(n, f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))} a(n)={polcoeff(lift(M(n, i->i^2)^2 * M(n, i->-(i^2)) ), 1%n)} -
PARI
a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, if(e>2, 2^(2*e+1), if(e==1, 4, 24)), (p+1)*p^(2*e-1)))}
Formula
Multiplicative with a(2^1) = 4, a(2^2) = 24, a(2^e) = 2^(2*e+1) for e > 2, a(p^e) = (p+1)*p^(2*e-1) for odd prime p.
a(n) = n^2*Sum_{d|n} mu(d)^2/d for n odd.
a(n) = A229179(n) for n mod 4 <> 0.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 19/(4*Pi^2) = 0.4812756... . - Amiram Eldar, Oct 18 2022