A308949 - OEIS

%I #15 Aug 10 2019 09:25:23

%S 1,2,5,3,29,7,169,17,197,41,5741,11,33461,239,269,577,1136689,199,

%T 6625109,1121,45697,8119,225058681,1153,45232349,47321,7761797,38081,

%U 44560482149,961,259717522849,665857,52734529,1607521,1800193921,13067,51422757785981

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

%C 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 a(n) = A008555(n) / gcd(A008555(n), n) if n != 2.

%e 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 nmax = 40;

%t pell = {1, 2};

%t pp = {1, 2};

%t Do[s = 2*pell[[-1]] + pell[[-2]];

%t   AppendTo[pell, s];

%t   AppendTo[pp, s/Times @@ pp[[Most[Divisors[n]]]]], {n, 3, nmax}];

%t a[2] = 2;

%t a[n_] := pp[[n]]/GCD[pp[[n]], n];

%t Array[a, nmax] (* _Jean-François Alcover_, Jul 06 2019, after _T. D. Noe_ in A008555 *)

%o (PARI) T(n) = ([2, 1; 1, 0]^n)[2, 1]

%o b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))

%o a(n) = if(n==2, 2, b(n)/gcd(n, b(n)))

%Y Cf. A000129, A008555, A178763, A214028, A238736.

%K nonn

%O 1,2

%A _Jianing Song_, Jul 02 2019