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A347277
Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.
0
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
1,4
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).
REFERENCES
W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
FORMULA
T(n,k) = (k^n - k^(n - phi(n)))/n.
EXAMPLE
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, ...
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);
PROG
(PARI) T(n, k) = (k^n - k^(n - eulerphi(n)))/n; \\ Jinyuan Wang, Aug 28 2021
CROSSREFS
Cf. A074650.
Sequence in context: A316781 A290733 A113020 * A357734 A228821 A336932
KEYWORD
nonn,tabl
AUTHOR
Franz Vrabec, Aug 26 2021
EXTENSIONS
More terms from Jinyuan Wang, Aug 28 2021
STATUS
approved