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

Author

Jonathan Vos Post, Nov 13 2012

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001358, A219117.

Programs

Magma

s:=func; [k : k in [2..2500]| forall{i:i in [1,2,3,4]| s(k^i+k+1)}]; // Marius A. Burtea, Feb 11 2020

PARI

is(n)=vector(4,i,bigomega(n^i+n+1))==[2,2,2,2] \\ Charles R Greathouse IV, Nov 13 2012