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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).

All primes are the terms of a(n). Quotients A130072(p)/p for p = Prime(n) are listed in A130075(n) = {6,30,570,10830,4422630,93776970,44871187170,1003806502230,...}. p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0. Nonprimes n such that n divides A130072(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.

Numbers k such that k divides A130072(k) 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,...}, which includes all primes. a(n) includes nonprimes in A130073(n). p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0. It appears that a(n) includes 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.

p divides 5^p - 3^p - 2^p = A130072(p) for prime p.

p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0.

2 divides a(n). 3 divides a(n). 5 divides a(n) for n>1. 19 divides a(n) for n>2. 19^2 divides a(n) for n in A091178(n) or prime(n) in A002476.

Are there only finitely many such numbers?

Primes in the sequence are A130076.

Next term, if it exists, exceeds 5*10^8. - Jon E. Schoenfield, May 07 2021