A137401 a(n) is the number of ordered solutions (x,y,z) to x^3 + y^3 == z^3 mod n with 1 <= x,y,z <= n-1.

0, 0, 2, 7, 12, 20, 0, 63, 116, 72, 90, 131, 0, 108, 182, 339, 240, 602, 324, 415, 326, 420, 462, 839, 604, 216, 1808, 763, 756, 812, 810, 1735, 992, 1056, 1092, 3311, 648, 1620, 650, 2511, 1560, 1640, 1134, 2227, 4328, 1980, 2070, 3683, 2484, 2644, 2450, 1519

Offset: 1

Author

Neven Juric (neven.juric(AT)apis-it.hr), Apr 11 2008

Keywords

• nonn

Comments

Record values of A137401: 0, 2, 7, 12, 20, 63, 116, 131, 182, 339, 602, 839, 1808, 3311, 4328, 7964, 8864, 9231, 19583, 21986, 41363, 52676, 81467, 87596, 92087, 112616, 236951, 247940, 378071, 386423, 521135, ... - Robert G. Wilson v

Examples

			a(4)=7 because (1, 2, 1), (1, 3, 2), (2, 1, 1), (2, 2, 2), (2, 3, 3), (3, 1, 2), (3, 2, 3) are solutions for n=4.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..425 from Robert G. Wilson v)

Crossrefs

Cf. A063454.

Programs

Mathematica

f[n_] := Block[ {c = 0}, Do[ If[ Mod[x^3 + y^3, n] == Mod[z^3, n], c++ ], {x, n - 1}, {y, n - 1}, {z, n - 1}]; c];
Table[Length[Select[Tuples[Range[n - 1], 3], Mod[ #[[1]]^3 + #[[2]]^3 - #[[3]]^3, n] == 0 &]], {n, 2, 50}] (* Stefan Steinerberger, Apr 12 2008 *)

Python

def A137401(n):
    ndict = {}
    for i in range(1,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

Formula

a(n) = A063454(n)-3*A087786(n)+3*A000189(n)-1. - Vladeta Jovovic, Apr 11 2008

Extensions

More terms from Stefan Steinerberger and Robert G. Wilson v, Apr 12 2008