A062775 -id:A062775 - cp's OEIS frontend

A063454 Number of solutions to x^3 + y^3 = z^3 mod n.

1, 4, 9, 20, 25, 36, 55, 112, 189, 100, 121, 180, 109, 220, 225, 448, 289, 756, 487, 500, 495, 484, 529, 1008, 725, 436, 2187, 1100, 841, 900, 1081, 2048, 1089, 1156, 1375, 3780, 973, 1948, 981, 2800, 1681, 1980, 1513, 2420, 4725, 2116, 2209, 4032

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 25 2001

Equivalently, the number of solutions to x^3 + y^3 + z^3 == 0 (mod n). - Andrew Howroyd, Jul 18 2018

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from Seiichi Manyama)

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), this sequence (k=3), A288099 (k=4), A288100 (k=5), A288101 (k=6), A288102 (k=7), A288103 (k=8), A288104 (k=9), A288105 (k=10).

a(n)={my(p=Mod(sum(i=0, n-1, x^(i^3%n)), 1-x^n)); polcoeff(lift(p^3), 0)} \\ Andrew Howroyd, Jul 18 2018

def A063454(n):
    ndict = {}
    for i in range(n):
        m = pow(i,3,n)
        if m in ndict:
            ndict[m] += 1
        else:
            ndict[m] = 1
    count = 0
    for i in ndict:
        ni = ndict[i]
        for j in ndict:
            k = (i+j) % n
            if k in ndict:
                count += ni*ndict[j]*ndict[k]
    return count # Chai Wah Wu, Jun 06 2017

More terms from Dean Hickerson, Jul 26 2001

A288099 Number of solutions to x^4 + y^4 = z^4 mod n.

Original entry on oeis.org

1, 4, 9, 24, 33, 36, 49, 192, 99, 132, 121, 216, 97, 196, 297, 1536, 193, 396, 361, 792, 441, 484, 529, 1728, 925, 388, 1377, 1176, 1121, 1188, 961, 6144, 1089, 772, 1617, 2376, 1441, 1444, 873, 6336, 481, 1764, 1849, 2904, 3267, 2116, 2209, 13824, 2695, 3700, 1737

Offset: 1

Seiichi Manyama, Jun 05 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), this sequence (k=4), A288100 (k=5), A288101 (k=6), A288102 (k=7), A288103 (k=8), A288104 (k=9), A288105 (k=10).

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^4)), 1-x^n)); vecsum(Vec( serconvol(lift(p^2) + O(x^n), lift(p) + O(x^n))))} \\ Andrew Howroyd, Jul 17 2018

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288100 Number of solutions to x^5 + y^5 = z^5 mod n.

Original entry on oeis.org

1, 4, 9, 20, 25, 36, 49, 112, 99, 100, 151, 180, 169, 196, 225, 704, 289, 396, 361, 500, 441, 604, 529, 1008, 1625, 676, 1377, 980, 841, 900, 1951, 4864, 1359, 1156, 1225, 1980, 1369, 1444, 1521, 2800, 601, 1764, 1849, 3020, 2475, 2116, 2209, 6336, 2695, 6500, 2601

Offset: 1

Seiichi Manyama, Jun 05 2017

Equivalently, the number of solutions to x^5 + y^5 + z^5 == 0 (mod n). - Andrew Howroyd, Jul 17 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), this sequence (k=5), A288101 (k=6), A288102 (k=7), A288103 (k=8), A288104 (k=9), A288105 (k=10).

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^5)), 1-x^n)); polcoeff(lift(p^3), 0)} \\ Andrew Howroyd, Jul 17 2018

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288101 Number of solutions to x^6 + y^6 = z^6 mod n.

Original entry on oeis.org

1, 4, 9, 24, 25, 36, 73, 192, 243, 100, 121, 216, 217, 292, 225, 1024, 289, 972, 217, 600, 657, 484, 529, 1728, 725, 868, 2673, 1752, 841, 900, 1441, 6144, 1089, 1156, 1825, 5832, 3241, 868, 1953, 4800, 1681, 2628, 505, 2904, 6075, 2116, 2209, 9216, 3871, 2900, 2601

Offset: 1

Seiichi Manyama, Jun 05 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), A288100 (k=5), this sequence (k=6), A288102 (k=7), A288103 (k=8), A288104 (k=9), A288105 (k=10).

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^6)), 1-x^n)); vecsum(Vec( serconvol(lift(p^2) + O(x^n), lift(p) + O(x^n))))} \\ Andrew Howroyd, Jul 17 2018

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288102 Number of solutions to x^7 + y^7 = z^7 mod n.

Original entry on oeis.org

1, 4, 9, 20, 25, 36, 49, 112, 99, 100, 121, 180, 169, 196, 225, 704, 289, 396, 361, 500, 441, 484, 529, 1008, 725, 676, 1377, 980, 589, 900, 961, 4864, 1089, 1156, 1225, 1980, 1369, 1444, 1521, 2800, 1681, 1764, 4999, 2420, 2475, 2116, 2209, 6336, 10633, 2900

Offset: 1

Seiichi Manyama, Jun 05 2017

Equivalently, the number of solutions to x^7 + y^7 + z^7 == 0 (mod n). - Andrew Howroyd, Jul 17 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), A288100 (k=5), A288101 (k=6), this sequence (k=7), A288103 (k=8), A288104 (k=9), A288105 (k=10).

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^7)), 1-x^n)); polcoeff(lift(p^3), 0)} \\ Andrew Howroyd, Jul 17 2018

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288103 Number of solutions to x^8 + y^8 = z^8 mod n.

Original entry on oeis.org

1, 4, 9, 24, 33, 36, 49, 192, 99, 132, 121, 216, 97, 196, 297, 1536, 385, 396, 361, 792, 441, 484, 529, 1728, 925, 388, 1377, 1176, 1121, 1188, 961, 12288, 1089, 1540, 1617, 2376, 1441, 1444, 873, 6336, 641, 1764, 1849, 2904, 3267, 2116, 2209, 13824, 2695, 3700

Offset: 1

Seiichi Manyama, Jun 05 2017

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), A288100 (k=5), A288101 (k=6), A288102 (k=7), this sequence (k=8), A288104 (k=9), A288105 (k=10).

Table[cnt=0; Do[If[Mod[x^8 + y^8 - z^8, n]==0, cnt++], {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 50}] (* Vincenzo Librandi, Jul 18 2018 *)

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^8)), 1-x^n)); vecsum(Vec( serconvol(lift(p^2) + O(x^n), lift(p) + O(x^n))))} \\ Andrew Howroyd, Jul 17 2018

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288104 Number of solutions to x^9 + y^9 = z^9 mod n.

Original entry on oeis.org

1, 4, 9, 20, 25, 36, 55, 112, 189, 100, 121, 180, 109, 220, 225, 704, 289, 756, 487, 500, 495, 484, 529, 1008, 725, 436, 5103, 1100, 841, 900, 1081, 4864, 1089, 1156, 1375, 3780, 973, 1948, 981, 2800, 1681, 1980, 1513, 2420, 4725, 2116, 2209, 6336, 2989, 2900, 2601

Offset: 1

Seiichi Manyama, Jun 05 2017

Equivalently, the number of solutions to x^9 + y^9 + z^9 == 0 (mod n). - Andrew Howroyd, Jul 17 2018

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), A288100 (k=5), A288101 (k=6), A288102 (k=7), A288103 (k=8), this sequence (k=9), A288105 (k=10).

Table[cnt=0; Do[If[Mod[x^9 + y^9 - z^9, n]==0, cnt++], {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 50}] (* Vincenzo Librandi, Jul 18 2018 *)

a(n)={my(p=Mod(sum(i=0, n-1, x^lift(Mod(i,n)^9)), 1-x^n)); polcoeff(lift(p^3), 0)} \\ Andrew Howroyd, Jul 17 2018

def A288104(n):
    ndict = {}
    for i in range(n):
        m = pow(i,9,n)
        if m in ndict:
            ndict[m] += 1
        else:
            ndict[m] = 1
    count = 0
    for i in ndict:
        ni = ndict[i]
        for j in ndict:
            k = (i+j) % n
            if k in ndict:
                count += ni*ndict[j]*ndict[k]
    return count # Chai Wah Wu, Jun 05 2017

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A288105 Number of solutions to x^10 + y^10 = z^10 mod n.

Original entry on oeis.org

1, 4, 9, 24, 25, 36, 49, 192, 99, 100, 201, 216, 169, 196, 225, 1024, 289, 396, 361, 600, 441, 804, 529, 1728, 3125, 676, 1377, 1176, 841, 900, 601, 6144, 1809, 1156, 1225, 2376, 1369, 1444, 1521, 4800, 1201, 1764, 1849, 4824, 2475, 2116, 2209, 9216, 2695, 12500

Offset: 1

Seiichi Manyama, Jun 05 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Number of solutions to x^k + y^k = z^k mod n: A062775 (k=2), A063454 (k=3), A288099 (k=4), A288100 (k=5), A288101 (k=6), A288102 (k=7), A288103 (k=8), A288104 (k=9), this sequence (k=10).

Table[cnt=0; Do[If[Mod[x^10 + y^10 - z^10, n]==0, cnt++], {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 50}] (* Vincenzo Librandi, Jul 18 2018 *)

def A288105(n):
    ndict = {}
    for i in range(n):
        m = pow(i,10,n)
        if m in ndict:
            ndict[m] += 1
        else:
            ndict[m] = 1
    count = 0
    for i in ndict:
        ni = ndict[i]
        for j in ndict:
            k = (i+j) % n
            if k in ndict:
                count += ni*ndict[j]*ndict[k]
    return count # Chai Wah Wu, Jun 05 2017

Keyword:mult added by Andrew Howroyd, Jul 17 2018

A087687 Number of solutions to x^2 + y^2 + z^2 == 0 (mod n).

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 32, 99, 100, 121, 72, 169, 196, 225, 64, 289, 396, 361, 200, 441, 484, 529, 288, 725, 676, 891, 392, 841, 900, 961, 256, 1089, 1156, 1225, 792, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 2475, 2116, 2209, 576, 2695, 2900, 2601

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

To show that a(n) is multiplicative is simple number theory. If gcd(n,m)=1, then any solution of x^2 + y^2 + z^2 == 0 (mod n) and any solution (mod m) can combined to a solution (mod nm) using the Chinese Remainder Theorem and any solution (mod nm) gives solutions (mod n) and (mod m). Hence a(nm) = a(n)*a(m). - Torleiv Kløve, Jan 26 2009

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 80 terms from Robert Gerbicz)
C. Calderón and M. J. De Velasco, On divisors of a quadratic form, Boletim da Sociedade Brasileira de Matemática, Vol. 31, No. 1 (2000), pp. 81-91; alternative link.
László Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq., Vol. 17 (2014), Article # 14.11.6; arXiv preprint, arXiv:1404.4214 [math.NT], 2014.
Index to sequences related to sums of squares

Cf. A086933, A062775.

Different from A064549.

A087687 := 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^(e+ceil(e/2)) ;
        elif type(e,'odd') then
            a := a*p^((3*e-1)/2)*(p^((e+1)/2)+p^((e-1)/2)-1) ;
        else
            a := a*p^(3*e/2-1)*(p^(e/2+1)+p^(e/2)-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A087687(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

a[n_] := Module[{k=1}, Do[{p, e} = pe; k = k*If[p == 2, p^(e + Ceiling[ e/2]), If[OddQ[e], p^((3*e-1)/2)*(p^((e+1)/2) + p^((e-1)/2) - 1), p^(3*e/2 - 1)*(p^(e/2 + 1) + p^(e/2) - 1)]], {pe, FactorInteger[n]}]; k];
Array[a, 100] (* Jean-François Alcover, Jul 10 2018, after R. J. Mathar *)

a(n)=local(v=vector(n),w);for(i=1,n,v[i^2%n+1]++);w=vector(n,i,sum(j=1,n,v[j]*v[(i-j)%n+1]));sum(j=1,n,w[j]*v[(1-j)%n+1]) \\ Martin Fuller

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 2^(e+(e+1)\2), p^(e+(e-1)\2)*(p^(e\2)*(p+1) - 1)))} \\ Andrew Howroyd, Aug 06 2018

a(2^k) = 2^(k + ceiling(k/2)). For odd primes p, a(p^(2k-1)) = p^(3k-2)*(p^k + p^(k-1) - 1) and a(p^(2k)) = p^(3k-1)*(p^(k+1) + p^k - 1). - Martin Fuller, Jan 26 2009

Sum_{k=1..n} a(k) ~ (4*zeta(3))/(15*zeta(4)) * n^3 + O(n^2 * log(n)) (Calderón and de Velasco, 2000). - Amiram Eldar, Mar 04 2021

More terms from Robert Gerbicz, Aug 22 2006

Edited by Steven Finch, Feb 06 2009, Feb 12 2009

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

cp's OEIS Frontend

A063454 Number of solutions to x^3 + y^3 = z^3 mod n.

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A288099 Number of solutions to x^4 + y^4 = z^4 mod n.

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A288100 Number of solutions to x^5 + y^5 = z^5 mod n.

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A288101 Number of solutions to x^6 + y^6 = z^6 mod n.

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A288102 Number of solutions to x^7 + y^7 = z^7 mod n.

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A288103 Number of solutions to x^8 + y^8 = z^8 mod n.

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A288104 Number of solutions to x^9 + y^9 = z^9 mod n.

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A288105 Number of solutions to x^10 + y^10 = z^10 mod n.

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