A191457 -id:A191457 - cp's OEIS frontend

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A191458 Primes p such that the polynomial x^2+x+p generates only primes for x=1..11.

Original entry on oeis.org

17, 41, 1761702947, 8776320587, 10102729577, 11085833111, 177558051107, 273373448057, 473787509537, 557149355507, 715464238661, 1359854730821, 2131528031441, 2170341748697, 2236159108277, 2308235320997, 2751203698151, 3247566894821, 3288002848511, 3424305123047

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Subsequence of A191457.

The sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Oct 11 2011

Charles R Greathouse IV and Zak Seidov, Table of n, a(n) for n = 1..79

Cf. A187060, A190800, A191456, A191457.

is(n)=for(x=0,11, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

a(12)-a(20) from Charles R Greathouse IV, Oct 17 2011

A247949 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.

Original entry on oeis.org

7, 43, 79, 457, 877, 967, 1093, 2437, 2683, 3187, 5077, 5923, 7933, 8233, 11923, 12889, 15787, 17389, 19993, 31543, 41113, 41617, 42457, 71359, 77863, 80683, 91393, 101719, 102643, 105967, 107347, 120163, 129733, 137593, 151783, 170263, 175723, 197569, 210127

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

All terms == 1 mod 6. - Robert Israel, Jan 11 2015

			a(1) = 7:
0^4 + 0^3 + 0^2 + 0 + 7 = 7;
1^4 + 1^3 + 1^2 + 1 + 7 = 11;
2^4 + 2^3 + 2^2 + 2 + 7 = 37;
3^4 + 3^3 + 3^2 + 3 + 7 = 127;
4^4 + 4^3 + 4^2 + 4 + 7 = 347;
5^4 + 5^3 + 5^2 + 5 + 7 = 787;
all six are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..6730

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

select(p -> andmap(isprime, [p, p+4, p+30, p+120, p+340, p+780]), [seq(6*i+1, i=1..10^5)]); # Robert Israel, Jan 11 2015

Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[20000]],AllTrue[#+{4,30,120,340,780},PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)

forprime(p=1, 500000, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780), print1(p,", ")))

A247966 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...6.

Original entry on oeis.org

43, 457, 967, 1093, 5923, 8233, 11923, 15787, 41113, 80683, 151783, 210127, 213943, 294919, 392737, 430879, 495559, 524827, 537007, 572629, 584557, 711727, 730633, 731593, 1097293, 1123879, 1138363, 1149163, 1396207, 1601503, 1739557, 1824139, 2198407, 2223853

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

			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;
all seven are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1405

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5, 6}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[200000]],AllTrue[#+{4,30,120,340,780,1554},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 10 2017 *)

forprime(p=1, 1e6, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)&  isprime(p+1554), print1(p,", ")))

A248206 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...7.

Original entry on oeis.org

43, 457, 967, 11923, 15787, 41113, 213943, 294919, 392737, 430879, 524827, 572629, 730633, 1097293, 1149163, 2349313, 2738779, 3316147, 3666007, 5248153, 5396617, 5477089, 7960009, 9949627, 10048117, 11260237, 11613289, 15281023, 16153279, 17250367, 18733807

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

			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;
all eight are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..511

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

Select[f=k^4+k^3+k^2+k; k={0,1,2,3,4,5,6,7}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[12*10^5]],AllTrue[#+{4,30,120,340,780,1554,2800},PrimeQ]&] (* Harvey P. Dale, Apr 24 2022 *)

forprime(p=1, 1e8, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)&  isprime(p+1554)& isprime(p+2800), print1(p,", ")))

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.

Original entry on oeis.org

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

Offset: 1

K. D. Bajpai, Jan 18 2015

nonn

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

			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.

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

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

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 *)

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,", ")))

Edited by Wolfdieter Lang, Feb 20 2015

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

cp's OEIS Frontend

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

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A191458 Primes p such that the polynomial x^2+x+p generates only primes for x=1..11.

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A247949 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.

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A247966 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...6.

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A248206 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...7.

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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.

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