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A308949 a(n) is the greatest divisor of A000129(n) that is coprime to A000129(m) for all positive integers m < n.

%I A308949 #15 Aug 10 2019 09:25:23
%S A308949 1,2,5,3,29,7,169,17,197,41,5741,11,33461,239,269,577,1136689,199,
%T A308949 6625109,1121,45697,8119,225058681,1153,45232349,47321,7761797,38081,
%U A308949 44560482149,961,259717522849,665857,52734529,1607521,1800193921,13067,51422757785981
%N A308949 a(n) is the greatest divisor of A000129(n) that is coprime to A000129(m) for all positive integers m < n.
%C A308949 a(n) is squarefree unless n is of the form A214028(A238736(k)) = {7, 30, 1546462, ...}. The terms in A238736 are called 2-Wall-Sun-Sun primes.
%F A308949 a(n) = A008555(n) / gcd(A008555(n), n) if n != 2.
%e A308949 A000129(30) = 107578520350 = 2 * 5^2 * 7 * 29 * 31^2 * 41 * 269. We have 2, 7 divides A000129(6) = 70; 29, 41 divides A000129(10) = 2378; 5, 269 divides A000129(15) = 195025, but A000129(m) is coprime to 31 for all 1 <= m < 30, so a(30) = 31^2 = 961.
%t A308949 nmax = 40;
%t A308949 pell = {1, 2};
%t A308949 pp = {1, 2};
%t A308949 Do[s = 2*pell[[-1]] + pell[[-2]];
%t A308949   AppendTo[pell, s];
%t A308949   AppendTo[pp, s/Times @@ pp[[Most[Divisors[n]]]]], {n, 3, nmax}];
%t A308949 a[2] = 2;
%t A308949 a[n_] := pp[[n]]/GCD[pp[[n]], n];
%t A308949 Array[a, nmax] (* _Jean-François Alcover_, Jul 06 2019, after _T. D. Noe_ in A008555 *)
%o A308949 (PARI) T(n) = ([2, 1; 1, 0]^n)[2, 1]
%o A308949 b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
%o A308949 a(n) = if(n==2, 2, b(n)/gcd(n, b(n)))
%Y A308949 Cf. A000129, A008555, A178763, A214028, A238736.
%K A308949 nonn
%O A308949 1,2
%A A308949 _Jianing Song_, Jul 02 2019