A219183 Numbers n such that n^1+n+1, n^2+n+1, n^3+n+1 and n^4+n+1 are all semiprime.

84, 92, 129, 132, 182, 185, 195, 201, 234, 255, 264, 327, 333, 356, 407, 444, 449, 528, 705, 732, 794, 795, 881, 980, 1079, 1095, 1115, 1126, 1241, 1253, 1302, 1431, 1479, 1496, 1574, 1772, 1781, 1799, 1805, 1874, 1922, 2052, 2067, 2316, 2352, 2381, 2420

Offset

1,1

Comments

This is to semiprimes A001358 what A219117 is to primes A000040. - Franklin T. Adams-Watters

From Robert Gerbicz: there is no n for which n^k+n+1 is semiprime for k=1,2,3,4,5. Proof: n^5+n+1 = (n^2+n+1)*(n^3-n^2+1), here n^2+n+1 is semiprime, so for n > 1, n^5+n+1 has at least 3 factors, hence not a semiprime.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

a(1) = 84 because 84^4 + 84 + 1 = 49787221 = 11 * 4526111; 84^3 + 84 + 1 = 592789 = 29 * 20441; 84^2 + 84 + 1 = 7141 = 37 * 193; 84^1 + 84 + 1 = 169 = 13^2.
3^4+3+1 = 85 = 5*17 is semiprime, but 3^3+3+1 = 321 is prime, so 3 is not in this sequence.
8^4+8+1 = 4105 = 5 * 821 is semiprime, but 8^3+8+1 = 521 is prime, so 8 is not in this sequence.
20^4+20+1 = 160021 = 17 * 9413 is semiprime, and 20^3+20+1 = 8021 = 13 * 617 is semiprime, but 20^2+20+1 = 421 is prime, so 20 is not in this sequence.

Prog

(PARI) is(n)=vector(4, i, bigomega(n^i+n+1))==[2, 2, 2, 2] \\ Charles R Greathouse IV, Nov 13 2012
(Magma) s:=func<n|&+[d[2]: d in Factorization(n)] eq 2>; [k : k in [2..2500]| forall{i:i in [1, 2, 3, 4]| s(k^i+k+1)}]; // Marius A. Burtea, Feb 11 2020

Crossrefs

Cf. A001358, A219117.

Sequence in context: A068405 A349760 A045569 * A289218 A329182 A113931

Adjacent sequences: A219180 A219181 A219182 * A219184 A219185 A219186

Keyword

nonn

Author

Jonathan Vos Post, Nov 13 2012

Status

approved