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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A134463 Values of k such that 5k^2 + 5k + 1 is a palindromic prime.

Original entry on oeis.org

1, 4, 5, 565, 475081, 4565455, 4639740, 4623988479, 5760242508141202328
Offset: 1

Views

Author

Alexander Adamchuk, Oct 26 2007

Keywords

Comments

Corresponding centered decagonal palindromic primes are 5k^2 + 5k + 1 = A134462 = {11, 101, 151, 1598951, 1128512158211, ...}. Note that the first 4 terms of A134463 are palindromic as well.
a(9) > 1414213562372. - Donovan Johnson, Feb 13 2011
a(10) > 14142135623730950488. - Patrick De Geest, May 29 2021

Links

Crossrefs

Cf. A134462 = Centered decagonal palindromic primes; or palindromic primes of the form 5k^2 + 5k + 1.
Cf. A002385 = Palindromic primes.
Cf. A062786 = Centered 10-gonal numbers.
Cf. A090562 = Primes of the form 5k^2 + 5k + 1.
Cf. A090563 = Values of k such that 5k^2 + 5k + 1 is a prime.

Programs

  • Mathematica
    Do[ f=5k^2+5k+1; If[ PrimeQ[f] && FromDigits[ Reverse[ IntegerDigits[ f ] ] ] == f, Print[ k ] ], {k, 1, 500000} ]

Extensions

a(6), a(7) from D. S. McNeil, Mar 02 2009
a(8) from Donovan Johnson, Feb 13 2011
a(9) from Patrick De Geest, May 29 2021

A175063 Primes p such that 5*p^2 + 5*p + 1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 43, 61, 71, 73, 79, 101, 107, 109, 127, 131, 137, 139, 149, 151, 167, 191, 193, 239, 263, 269, 283, 311, 347, 373, 379, 389, 401, 433, 439, 443, 461, 479, 499, 541, 587, 641, 643, 659, 701, 719, 733, 773, 797, 821, 883, 887, 907, 919
Offset: 1

Views

Author

Vincenzo Librandi, Dec 20 2009

Keywords

Links

Crossrefs

Cf. A000040, A090563.

Programs

  • Mathematica
    Select[Prime[Range[200]],PrimeQ[5#^2+5#+1]&]  (* Harvey P. Dale, Feb 25 2011 *)

Formula

A090563 INTERSECT A000040. - R. J. Mathar, Jul 06 2010

Extensions

Corrected title. - Joerg Arndt, Mar 11 2010

A090111 Values of k such that {P(k), P(k+1), ..., P(k+6)} are all prime numbers, where P(k) = 4*k^2 - 154*k + 1523.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 45, 53, 66, 67, 84, 129, 130, 131, 266, 328, 329, 1619, 1620, 2655, 2937, 7509, 7510, 18030, 29283, 29714, 29715, 37630, 42037, 44473, 45905
Offset: 1

Views

Author

Labos Elemer, Dec 30 2003

Keywords

Comments

The terms are arguments providing a sequence of 7 polynomially consecutive primes with respect to 4*x^2 - 154*x + 1523, a polynomial communicated by Rivera (2003).

Examples

			k = 1 provides {1373, 1231, 1097, 971, 853, 743, 641}, a 7-chain of primes.

Links

Crossrefs

Cf. A090101, A090102, A090107, A090108, A090109, A090110, A055561, A090563.

Programs

  • Mathematica
    Flatten[Position[Partition[Table[If[PrimeQ[4n^2-154n+1523],1,0],{n,46000}],7,1],{1,1,1,1,1,1,1}]] (* Harvey P. Dale, Mar 06 2015 *)
  • PARI
    isp(x) = isprime(4*x^2 - 154*x + 1523);
lista(kmax) = {my(v = vector(7, k, isp(k))); for(k = 8, kmax, if(vecprod(v) == 1, print1(k - 7, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

A090106 Values of k such that {P(k), P(k+1), ..., P(k+12)} are all prime numbers, where P(k) = k^2 + k + 41.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 219
Offset: 1

Views

Author

Labos Elemer, Dec 22 2003

Keywords

Comments

a(n) is the first argument providing 13 "polynomially consecutive" primes with respect to the polynomial x^2 + x + 41.
a(29) > 5*10^9, if it exists. - Amiram Eldar, Sep 27 2024

Examples

			k = 219: {P(219), ..., P(231)} = {48221, ..., 53633}, i.e., 13 consecutive integer values substituted to P(x) = x^2 + x + 41 polynomial, all provide primes. The "classical case" includes one single 41-chain of PC-primes, see A055561.

Crossrefs

Cf. A055561, A090101, A090102, A090562, A090563.

Programs

  • Mathematica
    Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 + k + 41], {k, 1, 1000}], 13, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)
  • PARI
    isp(x) = isprime(x^2 + x + 41);
lista(kmax) = {my(v = vector(13, k, isp(k))); for(k = 14, kmax, if(vecprod(v) == 1, print1(k - 13, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Extensions

2 wrong terms removed by Amiram Eldar, Sep 27 2024

A111292 Numbers n such that 6*n^2 + 6*n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 18, 19, 20, 22, 23, 30, 31, 32, 33, 35, 36, 38, 41, 42, 43, 45, 46, 47, 51, 55, 58, 60, 65, 67, 73, 74, 77, 78, 84, 86, 88, 93, 95, 97, 100, 101, 104, 106, 107, 109, 112, 117, 120, 123, 124, 126, 129, 130, 132, 134, 135, 137, 143, 148, 151
Offset: 1

Views

Author

Parthasarathy Nambi, Nov 01 2005

Keywords

Examples

			If n=43 then 6*n^2 + 6*n + 1 = 11353 (prime).

Crossrefs

Cf. A111251, A090563.

Programs

A173001 Values of n such that exactly one of 5n+1 and 5n^2+5n+1 is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 24, 25, 32, 35, 42, 43, 44, 48, 49, 50, 52, 55, 56, 60, 61, 63, 69, 70, 71, 73, 74, 79, 81, 82, 84, 85, 86, 87, 88, 92, 93, 94, 98, 101, 106, 107, 108, 109, 110, 112, 115, 117, 120, 123, 126, 127, 131, 132, 135, 137
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Feb 07 2010

Keywords

Crossrefs

Cf. A090563.

Extensions

56 inserted by R. J. Mathar, Mar 09 2010
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