A366419 - OEIS

%I #25 Nov 16 2023 05:40:58

%S 1,5,15,11,65,21,133,34,135,75,561,63,949,192,975,129,1921,165,3097,

%T 99,525,663,6095,231,1625,741,1215,576,8149,525,12121,513,8415,2091,

%U 8645,495,21349,3081,2535,363,34481,315,32551,1989,8775,5316,51935,903,6517,1875,28815,1062

%N Number of distinct integers of the form (x^n + y^n) mod n^3.

%C It is enough to take x,y from {0,1,...,n^2-1}.

%C 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.

%C 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).

%o (PARI) a(n) = #setbinop((x, y)->Mod(x, n^3)^n+Mod(y, n^3)^n, [0..n^2-1]); \\ _Michel Marcus_, Oct 14 2023

%o (Python)

%o def A366419(n):

%o     m = n**3

%o     return len({(pow(x,n,m)+pow(y,n,m))%m for x in range(n**2) for y in range(x+1)}) # _Chai Wah Wu_, Nov 13 2023

%Y Cf. A366418, A121278, A366420, A365100.

%K nonn

%O 1,2

%A _Albert Mukovskiy_, Oct 11 2023