A276919 Number of solutions to x^3 + y^3 + z^3 + t^3 == 1 (mod n) for 1 <= x, y, z, t <= n.

1, 8, 27, 64, 125, 216, 336, 512, 1296, 1000, 1331, 1728, 1794, 2688, 3375, 4096, 4913, 10368, 7410, 8000, 9072, 10648, 12167, 13824, 15625, 14352, 34992, 21504, 24389, 27000, 30225, 32768, 35937, 39304, 42000, 82944, 48396, 59280, 48438, 64000, 68921, 72576, 77529, 85184, 162000, 97336

Offset

1,2

Comments

It appears that a(n) = n^3 for n in A088232. See also A066498. - Michel Marcus, Oct 11 2016

Links

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

Mathematica

JJJ[4, n, lam] = Sum[If[Mod[a^3 + b^3 + c^3 + d^3, n] == Mod[lam, n], 1, 0], {d, 0, n - 1}, {a, 0, n - 1}, {b, 0, n - 1}, {c, 0 , n - 1}]; Table[JJJ[4, n, 1], {n, 1, 50}]

Prog

(PARI) a(n) = sum(x=1, n, sum(y=1, n, sum(z=1, n, sum(t=1, n, Mod(x, n)^3 + Mod(y, n)^3 + Mod(z, n)^3 + Mod(t, n)^3 == 1)))); \\ Michel Marcus, Oct 11 2016
(PARI) qperms(v) = {my(r=1, t); v = vecsort(v); for(i=1, #v-1, if(v[i]==v[i+1], t++, r*=binomial(i, t+1); t=0)); r*=binomial(#v, t+1)}
a(n) = {my(t=0); forvec(v=vector(4, i, [1, n]), if(sum(i=1, 4, Mod(v[i], n)^3)==1, print1(v", "); t+=qperms(v)), 1); t} \\ David A. Corneth, Oct 11 2016
(Python)
def A276919(n):
    ndict = {}
    for i in range(n):
        i3 = pow(i, 3, n)
        for j in range(i+1):
            j3 = pow(j, 3, n)
            m = (i3+j3) % n
            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 = (1-i) % n
        if j in ndict:
            count += ndict[i]*ndict[j]
    return count # Chai Wah Wu, Jun 06 2017

Crossrefs

Cf. A000189, A047726, A060839, A063454, A087412, A087786, A254073, A276920.

Sequence in context: A353621 A126200 A213491 * A076989 A270437 A259603

Adjacent sequences: A276916 A276917 A276918 * A276920 A276921 A276922

Keyword

nonn,mult

Author

José María Grau Ribas, Sep 22 2016

Status

approved