A219183 - OEIS

%I #28 Sep 08 2022 08:46:04

%S 84,92,129,132,182,185,195,201,234,255,264,327,333,356,407,444,449,

%T 528,705,732,794,795,881,980,1079,1095,1115,1126,1241,1253,1302,1431,

%U 1479,1496,1574,1772,1781,1799,1805,1874,1922,2052,2067,2316,2352,2381,2420

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

%C This is to semiprimes A001358 what A219117 is to primes A000040. - _Franklin T. Adams-Watters_

%C 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.

%H Charles R Greathouse IV, <a href="/A219183/b219183.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 3^4+3+1 = 85 = 5*17 is semiprime, but 3^3+3+1 = 321 is prime, so 3 is not in this sequence.

%e 8^4+8+1 = 4105 = 5 * 821 is semiprime, but 8^3+8+1 = 521 is prime, so 8 is not in this sequence.

%e 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.

%o (PARI) is(n)=vector(4,i,bigomega(n^i+n+1))==[2,2,2,2] \\ _Charles R Greathouse IV_, Nov 13 2012

%o (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

%Y Cf. A001358, A219117.

%K nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 13 2012