A253915 - OEIS

%I #45 Sep 11 2021 02:53:02

%S 43,967,11923,213943,2349313,3316147,30637567,33421159,39693817,

%T 49978447,105963769,143405887,148248949,153756073,156871549,172981279,

%U 187310803,196726693,203625283,211977523,220825453,268375879,350968543,357834283,414486697,427990369

%N Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields primes for k = 0..8, but not for k = 9.

%C All the terms in this sequence are congruent to 1 (mod 3).

%H Jon E. Schoenfield, <a href="/A253915/b253915.txt">Table of n, a(n) for n = 1..155</a> (terms < 2*10^10)

%e a(1) = 43:

%e 0^4 + 0^3 + 0^2 + 0 + 43 =   43;

%e 1^4 + 1^3 + 1^2 + 1 + 43 =   47;

%e 2^4 + 2^3 + 2^2 + 2 + 43 =   73;

%e 3^4 + 3^3 + 3^2 + 3 + 43 =  163;

%e 4^4 + 4^3 + 4^2 + 4 + 43 =  383;

%e 5^4 + 5^3 + 5^2 + 5 + 43 =  823;

%e 6^4 + 6^3 + 6^2 + 6 + 43 = 1597;

%e 7^4 + 7^3 + 7^2 + 7 + 43 = 2843;

%e 8^4 + 8^3 + 8^2 + 8 + 43 = 4723;

%e all nine are primes, and

%e 9^4 + 9^3 + 9^2 + 9 + 43 = 7423 = 13*571 is composite.

%e The next prime for p=43 appears for k=13, namely 30983.

%t Select[Prime[Range[118*10^5]],AllTrue[#+{0,4,30,120,340,780,1554,2800,4680},PrimeQ]&&CompositeQ[#+7380]&] (* _Harvey P. Dale_, Sep 10 2021 *)

%o (PARI) forprime(p=1, 1e10, if(isprime(p+4)&& isprime(p+30)&& isprime(p+120)&& isprime(p+340)&& isprime(p+780)&& isprime(p+1554)&& isprime(p+2800)&& isprime(p+4680) && !isprime(p+7380), print1(p,", ")))

%Y Cf. A027445, A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458, A247949, A247966, A248206.

%K nonn

%O 1,1

%A _K. D. Bajpai_, Jan 18 2015

%E Edited by _Wolfdieter Lang_, Feb 20 2015

%E Corrected and extended by _Harvey P. Dale_, Sep 10 2021