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

1, 2, 5, 3, 29, 7, 169, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 1136689, 199, 6625109, 1121, 45697, 8119, 225058681, 1153, 45232349, 47321, 7761797, 38081, 44560482149, 961, 259717522849, 665857, 52734529, 1607521, 1800193921, 13067, 51422757785981

Offset: 1

Author

Jianing Song, Jul 02 2019

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

nmax = 40;
pell = {1, 2};
pp = {1, 2};
Do[s = 2*pell[[-1]] + pell[[-2]];
  AppendTo[pell, s];
  AppendTo[pp, s/Times @@ pp[[Most[Divisors[n]]]]], {n, 3, nmax}];
a[2] = 2;
a[n_] := pp[[n]]/GCD[pp[[n]], n];
Array[a, nmax] (* Jean-François Alcover, Jul 06 2019, after T. D. Noe in A008555 *)

PARI

T(n) = ([2, 1; 1, 0]^n)[2, 1]
b(n) = my(v=divisors(n)); prod(i=1, #v, T(v[i])^moebius(n/v[i]))
a(n) = if(n==2, 2, b(n)/gcd(n, b(n)))

Formula

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