A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.

0, 1, 0, 2, 1, 0, 3, 3, 2, 0, 4, 6, 8, 3, 0, 5, 10, 20, 18, 6, 0, 6, 15, 40, 60, 48, 8, 0, 7, 21, 70, 150, 204, 108, 18, 0, 8, 28, 112, 315, 624, 640, 312, 30, 0, 9, 36, 168, 588, 1554, 2500, 2340, 810, 56, 0, 10, 45, 240, 1008, 3360, 7560, 11160, 8160, 2184, 96, 0

Offset: 1

Author

Franz Vrabec, Aug 26 2021

Keywords

• nonn
• tabl

Comments

The quotient T(n,k) = (k^n - k^(n-phi(n)))/n results from the generalization k^n == k^(n-phi(n)) (mod n) of Euler's theorem (see Sierpiński, p. 243).

The n-th row of the table is equal to the n-th row of A074650 iff n = p^j (p prime, j>=1).

Examples

			T(4,3) = (3^4 - 3^2)/4 = 18.
Square array starts:
  0, 1,   2,   3,    4,    5, ...
  0, 1,   3,   6,   10,   15, ...
  0, 2,   8,  20,   40,   70, ...
  0, 3,  18,  60,  150,  315, ...
  0, 6,  48, 204,  624, 1554, ...
  0, 8, 108, 640, 2500, 7560, ...
		

References

• W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

Crossrefs

Cf. A074650.

Programs

Maple

with(numtheory):
T:= proc(n, k) (k^n-k^(n-phi(n)))/n end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..11);

PARI

T(n,k) = (k^n - k^(n - eulerphi(n)))/n; \\ Jinyuan Wang, Aug 28 2021

Formula

T(n,k) = (k^n - k^(n - phi(n)))/n.

Extensions

More terms from Jinyuan Wang, Aug 28 2021