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A130072 a(n) = 5^n - 3^n - 2^n. 6

%I #6 May 07 2021 13:47:37

%S 0,12,90,528,2850,14832,75810,383808,1932930,9705552,48648930,

%T 243605088,1219100610,6098716272,30503196450,152544778368,

%U 762810181890,3814309582992,19072323542370,95363943807648,476826695752770,2384154405761712,11920834803510690

%N a(n) = 5^n - 3^n - 2^n.

%C p divides a(p) for prime p. Quotients a(p)/p for p = Prime(n) are listed in A130075(n) = {6,30,570,10830,4422630,93776970,44871187170,1003806502230,...}. p^(k+1) divides a(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0. Numbers n such that n divides a(n) are listed in A130073(n) = {1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,23,24,25,27,29,31,32,36,37,41,43,...}.

%C Nonprimes n such that n divides a(n) are listed in A130074(n) = {1,4,6,8,9,12,15,16,18,24,25,27,32,36,44,45,48,54,64,72,75,81,95,96,...} which apparently include all powers p^k of primes p = {2,3,5,19} for k>1 and all powers of numbers of the form 2^k*3^m, 3^k*5^m, 5^k*19^m. 2 divides a(m). 2^2 divides a(2m). 2^(k+3) divides a(2^k*m) for k>0 and m>1. 3^(k+1) divides a(3^k*m). 5^(k+1) divides a(5^k*(1+2m)). 7^(k+1) divides a(1+6*7^k*m). 11 divides a(1+10m) and a(4+10m). 11^(k+1) divides a(1+10*11^k*m) and a(84+10*11^k*m) for k>0. 13^(k+1) divides a(1+12*13^k*m). 17 divides a(1+16m) and a(14+16m).

%C 17^2 divides a(1+16*17m) and a(270+16*17m). 17^3 divides a(1+16*17^2*m) and a(542+16*17^2*m). 19 divides a(1+6m) and a(5+6m). 19^2 divides a(1+6m). 19^(k+2) divides a(1+6*19^k*m) and a(19^k*(1+6*m)) for k>0. 23^(k+1) divides a(1+22*23^k*m). 29^(k+1) divides a(1+28*29^k*m).

%C It appears that p^(k+1) divides a(1+(p-1)*p^k*m) for prime p>5 and integer k>=0 with exception for p = 19 where p^(k+2) divides a(1+(p-1)*p^k*m). For k = 1 and m = 1 it means that p^2 divides a(p^2-p+1) for prime p>5 and 19^3 divides a(19^2-19+1).

%F a(n) = 5^n - 3^n - 2^n.

%t Table[ 5^n-3^n-2^n, {n,1,30} ]

%Y Cf. A130073, A130074, A130075, A130076.

%K nonn

%O 1,2

%A _Alexander Adamchuk_, May 06 2007

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