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

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

Author

Rémi Guillaume, Apr 10 2022

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

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.

Programs

Mathematica

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

Python

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

Formula

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