A316985 - OEIS

%I #20 Oct 18 2022 07:27:27

%S 1,4,12,24,30,48,56,128,108,120,132,288,182,224,360,512,306,432,380,

%T 720,672,528,552,1536,750,728,972,1344,870,1440,992,2048,1584,1224,

%U 1680,2592,1406,1520,2184,3840,1722,2688,1892,3168,3240,2208,2256,6144,2744,3000

%N Number of solutions to x^2 + y^2 - z^2 == 1 (mod n).

%H Robert Israel, <a href="/A316985/b316985.txt">Table of n, a(n) for n = 1..10000</a>

%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014), Article 14.11.6.

%F 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.

%F a(n) = n^2*Sum_{d|n} mu(d)^2/d for n odd.

%F a(n) = A229179(n) for n mod 4 <> 0.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = 19/(4*Pi^2) = 0.4812756... . - _Amiram Eldar_, Oct 18 2022

%p g:= proc(t)

%p    if t[1]=2 then 2^(2*t[2]+1)

%p    else (t[1]+1)*t[1]^(2*t[2]-1)

%p    fi

%p end proc:

%p g([2,1]):= 4: g([2,2]):= 24:

%p seq(convert(map(g, ifactors(n)[2]),`*`),n=1..100); # _Robert Israel_, Jul 20 2018

%t 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 *)

%o (PARI)

%o M(n, f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}

%o a(n)={polcoeff(lift(M(n, i->i^2)^2 * M(n, i->-(i^2)) ), 1%n)}

%o (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)))}

%Y Cf. A087784, A227553, A229179.

%K nonn,mult

%O 1,2

%A _Andrew Howroyd_, Jul 18 2018