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

%I A347277 #16 Sep 29 2021 09:01:22
%S A347277 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,
%T A347277 70,150,204,108,18,0,8,28,112,315,624,640,312,30,0,9,36,168,588,1554,
%U A347277 2500,2340,810,56,0,10,45,240,1008,3360,7560,11160,8160,2184,96,0
%N A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.
%C A347277 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).
%C A347277 The n-th row of the table is equal to the n-th row of A074650 iff n = p^j (p prime, j>=1).
%D A347277 W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
%F A347277 T(n,k) = (k^n - k^(n - phi(n)))/n.
%e A347277 T(4,3) = (3^4 - 3^2)/4 = 18.
%e A347277 Square array starts:
%e A347277   0, 1,   2,   3,    4,    5, ...
%e A347277   0, 1,   3,   6,   10,   15, ...
%e A347277   0, 2,   8,  20,   40,   70, ...
%e A347277   0, 3,  18,  60,  150,  315, ...
%e A347277   0, 6,  48, 204,  624, 1554, ...
%e A347277   0, 8, 108, 640, 2500, 7560, ...
%p A347277 with(numtheory):
%p A347277 T:= proc(n, k) (k^n-k^(n-phi(n)))/n end:
%p A347277 seq(seq(T(i, 1+d-i), i=1..d), d=1..11);
%o A347277 (PARI) T(n,k) = (k^n - k^(n - eulerphi(n)))/n; \\ _Jinyuan Wang_, Aug 28 2021
%Y A347277 Cf. A074650.
%K A347277 nonn,tabl
%O A347277 1,4
%A A347277 _Franz Vrabec_, Aug 26 2021
%E A347277 More terms from _Jinyuan Wang_, Aug 28 2021