cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

K. D. Bajpai, Jan 11 2015

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

K. D. Bajpai, Jan 11 2015

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

K. D. Bajpai, Jan 18 2015

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • 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 *)
  • 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,", ")))

Extensions

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

Views

Author

M. F. Hasler, May 26 2016

Keywords

Comments

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

Links

Crossrefs

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

Programs

Formula

a(n) = (81 - prime(n))/2 for 1 < n < 17.
a(n) = A273770((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Extensions

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020
Previous Showing 11-14 of 14 results.

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