A102326 -id:A102326 - cp's OEIS frontend

A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

17, 19, 23, 31, 101, 103, 173, 179, 257, 263, 293, 353, 373, 431, 467, 491, 521, 563, 593, 619, 641, 677, 719, 739, 773, 821, 829, 857, 859, 863, 881, 929, 941, 953, 1031, 1051, 1087, 1091, 1109, 1129, 1229, 1297, 1327, 1399, 1433, 1487, 1489, 1499, 1583

Offset: 1

Labos Elemer, Jan 05 2005

nonn

			p = 17, 1 + p^3 = 1 + 4913 = 2*3*3*3*7*13, so the largest prime factor is 13 < p = 17.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A065091, A073501, A102326-A102330.

Mathematica
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<Ray Chandler, Jan 08 2005 *)
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A309562 Numbers k such that the largest prime divisor of k^4+1 is less than k.

Original entry on oeis.org

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

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

nonn

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

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

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

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

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

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

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

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

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A102325 Primes p such that the largest prime divisor of p^3 + 1 is less than p.

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

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