A309027 Prime powers of the form 12*c^2 + 4*c + 3, where c is an arbitrary integer.

3, 11, 19, 43, 59, 179, 211, 283, 563, 619, 739, 1163, 1499, 1979, 2083, 2411, 3011, 3539, 4259, 4723, 7603, 8011, 8219, 10211, 11411, 12163, 14011, 14563, 14843, 17483, 20011, 23059, 25579, 26699, 28619, 29803, 30203, 33923, 36083, 36523, 41539, 49411, 54139, 55219, 55763, 59083

Offset

1,1

Comments

It is conjectured that all terms are prime. See Leung et al. p. 12.

All terms up to 10^9 are prime.

Since the Diophantine equation 12*c^2 + 4*c + 3 = x^2 has no solution, all terms p^e have either e=1 or e>=3 and odd. Up to 10^24, all terms are prime. - Giovanni Resta, Jul 08 2019

It appears that these are the primes of A271723. - Bill McEachen, Aug 14 2021

Links

Table of n, a(n) for n=1..46.

Ka Hin Leung, Koji Momihara and Qing Xiang, A new family of Hadamard matrices of order 4(2q^2+1), arXiv:1907.02623 [math.CO], 2019. See p. 3.

Prog

(PARI) isok(n) = isprimepower(n) && issquare(3*n-8) && (d=sqrtint(3*n-8)) && ((frac((d-1)/6) == 0) || (frac((d+1)/6) == 0));

Crossrefs

Cf. A271723.

Sequence in context: A163183 A007520 A294912 * A213891 A336790 A163851

Adjacent sequences: A309024 A309025 A309026 * A309028 A309029 A309030

Keyword

nonn

Author

Michel Marcus, Jul 08 2019

Status

approved