A366420 Number of distinct integers of the form (x^n + y^n) mod n^4.

1, 9, 45, 35, 325, 95, 931, 259, 1215, 625, 6171, 627, 12337, 2210, 14625, 2049, 32657, 2435, 58843, 1683, 11025, 12105, 140185, 4883, 40625, 16055, 32805, 14586, 236321, 11875, 375751, 16385, 277695, 59245, 302575, 16071, 789913, 97475, 98865, 13107, 1413721, 9405, 1399693

Offset: 1

Author

Albert Mukovskiy, Oct 11 2023

Keywords

• nonn

Comments

It is enough to take x,y from {0,1,...,n^3-1}.

It appears that the number of distinct integers of the form x^(p^k) + y^(p^k) (mod (p^k)^m) for a prime p>2 and natural k is A121278(p)*p^(k-1)*p^(k*(m-2)) for m>1. For m=1 see A366418.

It appears that the number of distinct integers of the form x^n + y^n (mod n^m) for an odd n, m>1 is A121278(n)*n^(m-2).

Crossrefs

Cf. A366418, A121278, A366419, A365101.

Programs

PARI

a(n) = #setbinop((x, y)->Mod(x, n^4)^n+Mod(y, n^4)^n, [0..n^3-1]); \\ Michel Marcus, Oct 14 2023

Python

def A366420(n):
    m = n**4
    return len({(pow(x,n,m)+pow(y,n,m))%m for x in range(n**3) for y in range(x+1)}) # Chai Wah Wu, Nov 12 2023