Rémi Guillaume - cp's OEIS frontend

User: Rémi Guillaume

Rémi Guillaume has authored 3 sequences.

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

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.

A353004 Numbers k such that 2*k^2 + 29 is semiprime.

Original entry on oeis.org

29, 30, 32, 35, 39, 44, 50, 57, 58, 61, 63, 65, 72, 74, 76, 84, 87, 88, 89, 91, 92, 94, 95, 97, 99, 102, 107, 109, 113, 116, 118, 120, 122, 123, 125, 126, 127, 134, 138, 144, 145, 146, 147, 148, 149, 150, 153, 154, 156, 157, 163, 164, 165, 166, 169, 174, 175, 179, 180, 182, 183, 191, 194, 196, 200

Offset: 1

Rémi Guillaume, Apr 15 2022

nonn

The least positive k for which 2*k^2 + 29 is neither prime nor semiprime is k = 185, which gives 2*k^2 + 29 = 68479 = 31*47^2.

Includes 29*k if 58*k^2 + 1 is prime; Bunyakovsky's conjecture implies there are infinitely many of these. - Robert Israel, Jul 29 2025

			a(5) = 39; 2*39^2 + 29 = 3071 = 37*83 is semiprime.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A007642, whose first term not in this sequence is 185.

Cf. A241554, A352949, A353388.

filter:= proc(k) numtheory:-bigomega(2*k^2+29) = 2 end proc;
select(filter, [$1..1000]); # Robert Israel, Jul 29 2025

Select[Range[200], PrimeOmega[2*#^2 + 29] == 2 &] (* Amiram Eldar, Apr 15 2022 *)

isok(k) = bigomega(2*k^2+29) == 2; \\ Michel Marcus, Apr 15 2022

from sympy import primeomega
def semiprime(n): return primeomega(n) == 2
print([k for k in range(140) if semiprime(2*k**2+29)]) # Michael S. Branicky, Apr 15 2022

A352800 Numbers k such that 2*k^2 + 29 is prime.

Original entry on oeis.org

0, 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, 31, 33, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 73, 75, 77, 78, 79, 80, 81, 82

Offset: 1

Sean A. Irvine at the suggestion of Rémi Guillaume, Apr 03 2022

Robert Price, Table of n, a(n) for n = 1..10000
Adrien-Marie Legendre, Essai sur la théorie des nombres, Paris: Duprat, 1798, p. 10.

Cf. A007641.

Select[Range[0, 32260], PrimeQ[2 #^2 + 29] &] (* Robert Price, Apr 15 2025 *)

cp's OEIS Frontend

User: Rémi Guillaume

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A353004 Numbers k such that 2*k^2 + 29 is semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A352800 Numbers k such that 2*k^2 + 29 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica