A353388 -id:A353388 - cp's OEIS frontend

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

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

A241554 Semiprimes generated by the polynomial 2 * n^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

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

cp's OEIS Frontend

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

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