A187057 -id:A187057 - cp's OEIS frontend

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.

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

A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

Original entry on oeis.org

39, 38, 37, 35, 34, 32, 31, 29, 26, 25, 22, 20, 19, 17, 14, 12, 11, 12, 12, 12, 12, 16, 15, 12, 12, 13, 14, 13, 13, 14, 13, 13, 13, 13, 14, 14, 14, 16, 16, 16, 15, 15, 16, 16, 17

Offset: 2

M. F. Hasler, May 26 2016

nonn

See A273595 for further information and (cross)references.
From the initial values, the sequence seems strictly decreasing, with a(n+1) - a(n) = (prime(n+1) - prime(n))/2; however, this property does not persist beyond n = 16.
This is the subsequence of A273770 with indices n corresponding to odd primes 2n+1, see formula. - M. F. Hasler, Feb 17 2020

Index to entries related to primes produced by polynomials.

Cf. A002837, A007634, A005846, A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

A273597(n)=A273770(prime(n)\2) \\ changed Feb 17 2020

A273597(n,p=prime(n),q=A273756(p\2))={for(x=1,oo,isprime(q+p*x+x^2)||return(x))} \\ M. F. Hasler, Feb 17 2020

a(n) = (81 - prime(n))/2 for 1 < n < 17.

a(n) = A273770((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020

A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

Original entry on oeis.org

23, 53, 89, 263, 599, 1283, 1979, 3449, 5399, 5639, 11813, 14543, 41213, 42443, 44249, 47129, 55799, 57773, 65699, 74699, 75983, 79613, 84299, 87539, 88643, 88793, 88799, 113153, 115763, 126473, 143813, 148913, 150203, 160073, 163973, 167099, 176489, 178799, 178889, 209249

Offset: 1

Alexander Yutkin, Jun 15 2025

nonn

			p=23: 23+8=31, 23+14=37, 23+18=41, 23+20=43 —> prime quintuple: (23, 31, 37, 41, 43).

Cf. A000040.

Cf. A172454 [2, 4, 6], A078855 [6, 4, 2], A187057 [2, 4, 6, 8].

[p: p in PrimesUpTo(300000) | IsPrime(p+8) and IsPrime(p+14) and IsPrime(p+18) and IsPrime(p+20)]; // Vincenzo Librandi, Jul 04 2025

q:= p-> andmap(i-> isprime(p+i), [0, 8, 14, 18, 20]):
select(q, [5+6*i$i=0..35000])[];  # Alois P. Heinz, Jun 16 2025

Select[Prime[Range[20000]], AllTrue[#+{8, 14, 18,20}, PrimeQ]&] (* Stefano Spezia, Jun 18 2025 *)

A385824 Primes p such that p + 10, p + 18, p + 24, p + 28 and p + 30 are also primes.

Original entry on oeis.org

13, 43, 79, 14533, 41203, 42433, 47119, 88789, 113143, 150193, 340909, 348433, 416389, 556243, 576193, 609589, 626599, 637699, 669649, 715849, 752263, 855709, 859249, 891799, 1107763, 1146763, 1189603, 1191079, 1201999, 1210369, 1225099, 1416043, 1510189, 1601599, 1893163

Offset: 1

Alexander Yutkin, Jul 09 2025

nonn

Initial members of prime sextuples that correspond to the difference pattern [10, 8, 6, 4, 2]. The primes in a sextuple do not have to be consecutive.

			p=13: 13+10=23, 13+18=31, 13+24=37, 13+28=41, 13+30=43 —> prime sextuple: (13, 23, 31, 37, 41, 43).

Cf. A000040.

Cf. A187057 [2, 4, 6, 8], A385035 [8, 6, 4, 2], A187058 [2, 4, 6, 8, 10].

Select[Prime[Range[150000]], And @@ PrimeQ[# + {10, 18, 24, 28, 30}] &] (* Amiram Eldar, Jul 09 2025 *)

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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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A273597 min { x >= 0 | A273595(n) + prime(n)*x + x^2 is composite }, where A273595(n) is such that this is a local maximum.

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A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

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A385824 Primes p such that p + 10, p + 18, p + 24, p + 28 and p + 30 are also primes.

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