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

0, 12, 90, 528, 2850, 14832, 75810, 383808, 1932930, 9705552, 48648930, 243605088, 1219100610, 6098716272, 30503196450, 152544778368, 762810181890, 3814309582992, 19072323542370, 95363943807648, 476826695752770, 2384154405761712, 11920834803510690

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_Alexander Adamchuk (alex(AT)kolmogorov.com), _, May 06 2007

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).

nonn,new

nonn

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,...}.

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,...}. 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). 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). 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).

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).

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).

nonn,new

nonn

5^n - 3^n - 2^n.

0, 12, 90, 528, 2850, 14832, 75810, 383808, 1932930, 9705552, 48648930, 243605088, 1219100610, 6098716272, 30503196450, 152544778368, 762810181890, 3814309582992, 19072323542370, 95363943807648, 476826695752770

1,2

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,...}. 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). 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). 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).

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

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

Cf. A130073, A130074, A130075, A130076.

nonn,new

Alexander Adamchuk (alex(AT)kolmogorov.com), May 06 2007

approved