A353004 -id:A353004 - cp's OEIS frontend

A241554 Semiprimes generated by the polynomial 2 * n^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

K. D. Bajpai, Apr 25 2014

nonn

2 * n^2 + 29 is a well-known Legendre prime-producing polynomial which generates 29 distinct primes for n = 0, 1, ..., 28. For n = 29, it yields the first semiprime, 1711 = 29 * 59.

The number n = 185 is the least positive integer for which 2*n^2 + 29 = 68479 = 31 * 47 * 47 is not squarefree.

			2 * 30^2 + 29 = 1829 = 31 * 59, which is a semiprime and is a term.
2 * 35^2 + 29 = 2479 = 37 * 67, which is a semiprime and is a term.

K. D. Bajpai, Table of n, a(n) for n = 1..10000
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A145292, A228183, A237627, A353004, A353388.

with(numtheory):A241554:= proc() local k; k:=2*x^2+29;if bigomega(k)=2 then RETURN (k); fi; end: seq(A241554(), x=0..500);

A241554 = {}; Do[k = 2 * n^2 + 29; If[PrimeOmega[k] == 2, AppendTo[A241554, k]], {n,200}]; A241554

s=[]; for(n=1, 200, t=2*n^2+29; if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A353388 Numbers k such that 2*k^2 + 29 is neither a prime nor a semiprime.

Original entry on oeis.org

185, 187, 232, 247, 261, 309, 311, 370, 371, 373, 435, 442, 464, 479, 501, 516, 520, 553, 557, 561, 590, 614, 619, 620, 621, 627, 638, 667, 701, 702, 705, 708, 714, 738, 755, 769, 796, 797, 802, 812, 836, 849, 853, 856, 869, 874, 890, 896, 899, 903, 906, 915, 943, 957, 960, 964, 973, 990

Offset: 1

Hugo Pfoertner, Apr 16 2022

nonn

If k is a term, then so is k + j*(2*k^2+29) for all natural numbers j. - Robert Israel, Jul 23 2023

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

Cf. A007642, A353004.

select(k -> numtheory:-bigomega(2*k^2+29) > 2, [$1..1000]); # Robert Israel, Jul 23 2023

Select[Range[1000], PrimeOmega[2*#^2 + 29] >= 3 &] (* Amiram Eldar, Apr 17 2022 *)

for(k=0,1000,if(bigomega(2*k^2+29) >= 3,print1(k,", ")))

from sympy import primeomega
def ok(n): return primeomega(2*n**2 + 29) >= 3
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, Apr 16 2022

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

Original entry on oeis.org

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.

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A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

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A353388 Numbers k such that 2*k^2 + 29 is neither a prime nor a semiprime.

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

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