A247949 -id:A247949

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A253915

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.

+10
2

43, 967, 11923, 213943, 2349313, 3316147, 30637567, 33421159, 39693817, 49978447, 105963769, 143405887, 148248949, 153756073, 156871549, 172981279, 187310803, 196726693, 203625283, 211977523, 220825453, 268375879, 350968543, 357834283, 414486697, 427990369

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..155 (terms < 2*10^10)

EXAMPLE

a(1) = 43:

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

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

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

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

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

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

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

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

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

all nine are primes, and

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

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

MATHEMATICA

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 *)

PROG

(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, ", ")))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

K. D. Bajpai, Jan 18 2015

EXTENSIONS

Edited by Wolfdieter Lang, Feb 20 2015

Corrected and extended by Harvey P. Dale, Sep 10 2021

STATUS

approved

A347537

a(n) is the smallest prime p such that the polynomial k^4 + k^3 + k^2 + k + p yields primes for k = 0..n-1, but not for k = n.

+10
0

2, 3, 13, 37, 109, 7, 1093, 457, 43, 430879, 130901527, 1838420599, 48181700197

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..13.

EXAMPLE

At k=0, k^4 + k^3 + k^2 + k + p is, of course, prime for every prime p.

a(1)=2 because 2 is the smallest prime p such that 1^4 + 1^3 + 1^2 + 1 + p = 4 + p is not prime: 4 + 2 = 6 = 2*3.

a(2)=3 because 3 is the smallest prime p such that k^4 + k^3 + k^2 + k + p is prime for k=1 but not for k=2, i.e., such that 4 + p is prime but 2^4 + 2^3 + 2^2 + 2 + p = 30 + p is not prime: 4 + 3 = 7 is prime but 30 + 3 = 33 = 3*11.

a(6)=7 because 7 is the smallest prime p such that k^4 + k^3 + k^2 + k + p is prime for k = 1..5, but not for k = 6: 4 + 7 = 11, 30 + 7 = 37, 120 + 7 = 127, 340 + 7 = 347, and 780 + 7 = 787, but 1554 + 7 = 1561 = 7*223.

CROSSREFS

Cf. A247949, A247966, A248206, A253915.

KEYWORD

nonn,more

AUTHOR

Jon E. Schoenfield, Sep 11 2021

EXTENSIONS

a(13) from Jinyuan Wang, Sep 11 2021

STATUS

approved

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