A240547 -id:A240547 - cp's OEIS frontend

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

1, 10, 33, 88, 145, 330, 385, 736, 945, 1450, 1441, 2904, 2353, 3850, 4785, 6016, 5185, 9450, 7201, 12760, 12705, 14410, 12673, 24288, 18625, 23530, 26001, 33880, 25201, 47850, 30721, 48640, 47553, 51850, 55825, 83160, 51985, 72010, 77649, 106720

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005353, A059306, A062801, A069097, A102631, A240547.

f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2020 *)

a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])

a(n)=local(c=0); forvec(x=vector(4,k,[1,n]),c+=((x[1]*x[2]-x[3]*x[4])%n==0)); c

From Vladeta Jovovic, Apr 22 2002: (Start)
a(n) = n^4 - A005353(n).
Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)
Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).
A102631(n) | a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
From Piotr Rysinski, Sep 11 2020: (Start)
a(n) = n * A069097(n).
Proof: a(n) is multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1), A069097(n) is multiplicative with A069097(p^e) = p^(e-1)*(p^e*(p+1)-1), so a(p^e) = p^e*A069097(p^e). (End)

A096018 Number of Pythagorean quadruples mod n; i.e., number of solutions to w^2 + x^2 + y^2 = z^2 mod n.

Original entry on oeis.org

1, 8, 21, 64, 145, 168, 301, 512, 621, 1160, 1221, 1344, 2353, 2408, 3045, 4096, 5185, 4968, 6517, 9280, 6321, 9768, 11661, 10752, 18625, 18824, 16281, 19264, 25201, 24360, 28861, 32768, 25641, 41480, 43645, 39744, 51985, 52136, 49413, 74240

Offset: 1

T. D. Noe, Jun 15 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..127 from R. J. Mathar, terms 128..2500 from Andrew Howroyd)
A. H. Hakami, Small zeros of quadratic congruences to a prime power modulus, PhD Thesis (2009), Lemma 4.4.
A. H. Hakami, Small primitive zeros of quadratic forms mod p^m, Raman. J. 38 (2015) 189-198, Lemma 2.1 for n=4, det Q=-1, omega_j(y')= p^(m-j)-p^(m-j-1).
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.
Index to sequences related to sums of squares.

Cf. A062775 (number of solutions to x^2 + y^2 = z^2 mod n), A240547.

Cf. A088539, A243381, A334425, A334426.

A096018 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(3*e) ;
        elif modp(p,4) = 1 then
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        else
            if type(e,'even') then
                a := a* (p^(3*e)+(p-1)*p^(2*e-1)*(1-p^e)/(1+p)) ;
            else
                a := a* (p^(3*e)-(p-1)*p^(2*e-1)*(1+p^e)/(1+p)) ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A096018(n),n=1..50) ; # R. J. Mathar, Jun 24 2018

Table[cnt=0; Do[If[Mod[w^2+x^2+y^2-z^2, n]==0, cnt++ ], {w, 0, n-1}, {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 50}]
f[2, e_] := 2^(3*e); f[p_, e_] := If[Mod[p, 4] == 1, p^(2*e - 1)*(p^(e + 1) + p^e - 1), If[EvenQ[e], p^(3*e) + (p - 1)*p^(2*e - 1)*(1 - p^e)/(1 + p), p^(3*e) - (p - 1)*p^(2*e - 1)*(1 + p^e)/(1 + p)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

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)^3 * M(n, i->-(i^2))), 0)} \\ Andrew Howroyd, Jun 23 2018

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, 2^(3*e), if(p%4 == 1, p^(2*e-1)*(p^(e+1) + p^e - 1), if(e%2, p^(3*e) - (p - 1)*p^(2*e - 1)*(1 + p^e)/(1 + p), p^(3*e) + (p - 1)*p^(2*e - 1)*(1 - p^e)/(1 + p)))));} \\ Amiram Eldar, Nov 21 2023

a(n) is multiplicative. For the powers of primes p, there are several cases. For p=2, we have a(2^e) = 2^(3e). For odd primes p with p==1 (mod 4), we have a(p^e) = p^(2*e-1)*(p^(e+1)+p^e-1). For odd primes p with p==3 (mod 4) and even e we have a(p^e) = p^(3*e) +(p-1)*p^(2*e-1)*(1-p^e)/(1+p). For odd primes p == 3 (mod 4) and odd e we have a(p^e) = p^(3*e) -(p-1)*p^(2*e-1)*(1+p^e)/(1+p). [Corrected Jun 24 2018, R. J. Mathar]

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A334425 * A334426 /(A088539 * A243381) = 0.94532146880744347512... . - Amiram Eldar, Nov 21 2023

A229294 Number of solutions to x^2 + y^2 + z^2 + t^2 == n (mod 2n) for x,y,z,t in [0, 2n).

Original entry on oeis.org

8, 96, 264, 384, 1160, 3168, 3080, 1536, 7560, 13920, 11528, 12672, 18824, 36960, 38280, 6144, 41480, 90720, 57608, 55680, 101640, 138336, 101384, 50688, 149000, 225888, 208008, 147840, 201608, 459360, 245768, 24576, 380424, 497760, 446600, 362880, 415880

Offset: 1

José María Grau Ribas, Sep 22 2013

nonn

All values are divisible by a(1)=8 and the sequence a(n)/8 is multiplicative. - Andrew Howroyd, Aug 07 2018

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A229295, A229296, A229297, A240547.

A[n_] := Sum[If[Mod[a^2 + b^2 + c^2 + d^2, 2n] == n, 1, 0], {d, 0, 2n - 1}, {a, 0, 2n - 1}, {b, 0, 2n - 1}, {c, 0, 2n - 1}];Table[Print[aaa = A[n]]; aaa, {n, 1, 40}]

a(n)={my(m=2*n); my(p=Mod(sum(i=0, m-1, x^(i^2%m)), x^m-1)^4); polcoeff( lift(p), n)} \\ Andrew Howroyd, Aug 07 2018

a(n)={my(f=factor(n)); 8*prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 3*2^(2*e), p^(2*e-1)*(p^(e+1)+p^e-1)))} \\ Andrew Howroyd, Aug 07 2018

def A229294(n):
    ndict = {}
    n2 = 2*n
    for i in range(n2):
        i3 = pow(i,2,n2)
        for j in range(i+1):
            j3 = pow(j,2,n2)
            m = (i3+j3) % n2
            if m in ndict:
                if i == j:
                    ndict[m] += 1
                else:
                    ndict[m] += 2
            else:
                if i == j:
                    ndict[m] = 1
                else:
                    ndict[m] = 2
    count = 0
    for i in ndict:
        j = (n-i) % n2
        if j in ndict:
            count += ndict[i]*ndict[j]
    return count # Chai Wah Wu, Jun 07 2017

a(n) = 8*A240547(n) for odd n, a(2^k) = 24*2^(2*k). - Andrew Howroyd, Aug 07 2018

A316148 Number of non-congruent solutions of x^2+y^2 == z^2+w^2 (mod n).

Original entry on oeis.org

1, 8, 33, 96, 145, 264, 385, 896, 945, 1160, 1441, 3168, 2353, 3080, 4785, 7680, 5185, 7560, 7201, 13920, 12705, 11528, 12673, 29568, 18625, 18824, 26001, 36960, 25201, 38280, 30721, 63488, 47553, 41480, 55825, 90720, 51985, 57608, 77649, 129920, 70561, 101640, 81313

Offset: 1

R. J. Mathar, Jun 25 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to sums of squares.

Cf. A096018, A240547, A306633.

A316148 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(2*e+1)*(p^e-1) ;
        else
            a := a*p^(2*e-1)*(p^(e+1)+p^e-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A316148(n),n=1..100) ;

f[2, e_] :=  2^(2*e+1)*(2^e-1); f[p_, e_] := p^(3*e)+p^(3*e-1)-p^(2*e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(2*e+1)*(2^e-1), p^(3*e)+p^(3*e-1)-p^(2*e-1)));} \\ Amiram Eldar, Dec 18 2023

Multiplicative with a(2^e) = 2^(2e+1)*(2^e-1), a(p^e) = p^(3e)+p^(3e-1)-p^(2e-1) for odd primes p.

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Dec 18 2023

cp's OEIS Frontend

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

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A096018 Number of Pythagorean quadruples mod n; i.e., number of solutions to w^2 + x^2 + y^2 = z^2 mod n.

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A229294 Number of solutions to x^2 + y^2 + z^2 + t^2 == n (mod 2*n) for x,y,z,t in [0, 2*n).

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A316148 Number of non-congruent solutions of x^2+y^2 == z^2+w^2 (mod n).

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A229294 Number of solutions to x^2 + y^2 + z^2 + t^2 == n (mod 2n) for x,y,z,t in [0, 2n).