A309562 Numbers k such that the largest prime divisor of k^4+1 is less than k.

1600, 2949, 3370, 8651, 8758, 8777, 9308, 9647, 10181, 10566, 10820, 11518, 12400, 12461, 13360, 13724, 14051, 14273, 14971, 16802, 18073, 18283, 18324, 18979, 22143, 22812, 23343, 23766, 24590, 24780, 25152, 25253, 25313, 25897, 26097, 26659, 27106, 27134, 28523, 28526, 29586, 29588, 30660

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 07 2019

Keywords

• nonn

Comments

To see if some m is a term we don't have to factor m^4 + 1 entirely. All we need to know is if the largest prime factor is less than k = m^4 + 1. - David A. Corneth, Jul 31 2020

Examples

			1600 is a member because 1600^4+1=17^2*113*337*641*929 has all its prime divisors < 1600.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..7762 (first 1000 terms from Robert Israel)

Crossrefs

Cf. A102326 (primes in this sequence), A256011.

Programs

Magma

[k: k in [1..31000]| Max(PrimeDivisors(k^4+1)) lt k]; // Marius A. Burtea, Aug 07 2019

Maple

filter := proc(n) max(numtheory:-factorset(n^4 + 1)) < n; end proc:
select(filter, [$1..40000]);

Mathematica

filterQ[k_] := FactorInteger[k^4 + 1][[-1, 1]] < k;
Select[Range[40000], filterQ] (* Jean-François Alcover, Jul 31 2020 *)

PARI

is(n) = my(f = factor(n^4 + 1, n + 1)); f[#f~, 1] < n \\ David A. Corneth, Jul 31 2020