A352800 -id:A352800 - cp's OEIS frontend

A007641 Primes of the form 2*k^2 + 29.

29, 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597, 1951, 2207, 2341, 2621, 2767, 2917, 3229, 3391, 3557, 3727, 4079, 4261, 4447, 4637, 4831, 5231, 5437

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

The first 29 terms of 2*k^2 + 29 (k = 0 to 28) are primes. This was discovered by Adrien-Marie Legendre. The sequence and its first 8 terms appear in the novel Code to Zero by Ken Follett. - Amiram Eldar, Apr 08 2017

Let P(k) = 2*k^2 + 29. The polynomial P(2*k - 28) = 8*k^2 - 224*k + 1597 produces prime values (not distinct) for k = 0 to 28. The polynomial P(3*k - 55) = 18*k^2 - 660*k + 6079 produces distinct prime values for k = 0 to 27. Cf. A050265. - Peter Bala, Apr 16 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ken Follett, Code to Zero, New York: Signet, 2001, p. 18.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Adrien-Marie Legendre, Essai sur la théorie des nombres, Paris: Duprat, 1798, p. 10.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Cf. A005846, A050265, A352800.

[a: n in [0..60] | IsPrime(a) where a is 2*n^2+29]; // Vincenzo Librandi, Mar 20 2013

Select[Table[2 n^2 + 29, {n, 0, 70}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

list(lim)=my(v=List(),t); for(n=0,sqrtint((lim-29)\2), if(isprime(t=2*n^2+29), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jan 20 2022

Edited by Erich Friedman, Feb 09 2002

A352949 Composite numbers of the form 2*k^2 + 29.

Original entry on oeis.org

1711, 1829, 2077, 2479, 3071, 3901, 5029, 6527, 6757, 7471, 7967, 8479, 10397, 10981, 11581, 14141, 15167, 15517, 15871, 16591, 16957, 17701, 18079, 18847, 19631, 20837, 22927, 23791, 25567, 26941, 27877, 28829, 29797, 30287, 31279, 31781, 32287, 35941, 38117

Offset: 1

Rémi Guillaume, Apr 10 2022

nonn

The first two terms that are not semiprimes, and their prime factorizations, are:
  a(62) = 2*185^2 + 29 = 68479 = 31*47*47,
  a(63) = 2*187^2 + 29 = 69967 = 31*37*61.
--
No number of the form 2^k*2 + 29 has any prime factor < 29, as can be proved by showing that 2*k^2 + 29 (mod p) takes only nonzero values for all primes p < 29:
  +----+-----------------------------------------------+
  |  p | Residues modulo p of 2*k^2 + 29               |
  +----+-----------------------------------------------+
  |  2 | 1                                             |
  |  3 | 1,  2                                         |
  |  5 | 1,  2,  4                                     |
  |  7 | 1,  2,  3,  5                                 |
  | 11 | 2,  3,  4,  6,  7,  9                         |
  | 13 | 1,  3,  5,  8,  9, 10, 11                     |
  | 17 | 3,  4,  8, 10, 11, 12, 13, 14, 16             |
  | 19 | 1,  3,  4,  5,  6,  9, 10, 12, 13, 18         |
  | 23 | 1,  6,  7,  8,  9, 10, 12, 14, 15, 18, 19, 22 |
  +----+-----------------------------------------------+
Idea and table from Jon E. Schoenfield.
Example of explanation:
if k ~ 0 (mod 3) then k^2 ~ 0 (mod 3), so 2*k^2 + 29 ~ 29 (mod 3) ~ 2 (mod 3);
if k ~ 1 (mod 3) or if k ~ 2 (mod 3) ~ -1 (mod 3), then k^2 ~ 1 (mod 3), so 2*k^2 + 29 ~ 31 (mod 3) ~ 1 (mod 3).
--
A number of the form 2*k^2 + 29 has the prime 29 as a factor iff k ~ 0 (mod 29).

			a(5) = 3071 = 37*83 = 2*39^2 + 29 is composite and of the form 2*k^2 + 29.
a(62) = 68479 = 31*47^2 = 2*185^2 + 29 is composite and of the form 2*k^2 + 29.

Michael S. Branicky, Table of n, a(n) for n = 1..10000.
Rémi Guillaume, Examples of prime factorizations and prime factor distributions, and proofs.

Cf. A007642 for arguments k.
Cf. 2*A353004^2 + 29 = A241554, which is a subsequence, for semiprimes.
Cf. 2*A352800^2 + 29 = A007641 for primes.

Select[2*Range[150]^2 + 29, CompositeQ] (* Amiram Eldar, Apr 15 2022 *)

from sympy import isprime
print([m for m in (2*k**2+29 for k in range(140)) if not isprime(m)]) # Michael S. Branicky, Apr 15 2022

a(n) = 2*(A007642(n))^2 + 29.

cp's OEIS Frontend

A007641 Primes of the form 2*k^2 + 29.

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