A240583 Odd primes satisfying a specific condition (see comments).

47, 79, 167, 191, 223, 239, 263, 359, 367, 383, 431, 439, 463, 479, 503, 599, 607, 719, 823, 839, 863, 887, 911, 983, 1031, 1039, 1087, 1103, 1223, 1231, 1303, 1319, 1327, 1367, 1399, 1439, 1447, 1487, 1511, 1543, 1559, 1583, 1663, 1759, 1823, 1831, 1847

Offset

1,1

Comments

Condition on odd prime p so that Q(Cp) is not rational over Q: p = 2q+1 where q = -1 (mod 4), q is squarefree, and any of 4p-q and q+1 is not square.

Consider the even integers E > a(1), and constrain distance <= E. The absolute distance from E to one of the sequence terms is conjectured to nearly always be prime (or adjacent). For example, 1488768 = a(40046)-1488623. There are four early exceptions thru E = 1488694, {104, 128, 134, 212}. Where is the next exception? - Bill McEachen, May 31 2025

Links

Table of n, a(n) for n=1..47.

Shizuo Endo and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan, Volume 25, Number 1 (1973), 1-167 (see Proposition 3.6 (i) p.18).

Akinari Hoshi, On Noether's problem for cyclic groups of prime order, arXiv:1402.3678 [math.NT], 2014 (see Proposition 3.1 (i) p.4 and Table 1 p.19).

Mathematica

Select[Prime[Range[2, 300]], (q = (#-1)/2; SquareFreeQ[q] && Mod[q, 4] == 3 && !IntegerQ[Sqrt[4#-q]] && !IntegerQ[Sqrt[q+1]])&] (* Jean-François Alcover, Sep 22 2018 *)

Prog

(PARI) isok(p) = isprime(p) && (q = (p-1)/2) && issquarefree(q) && ((q % 4) == 3) && (!issquare(4*p-q)) && (!issquare(q+1)); \\ corrected by Michel Marcus, May 31 2025

Crossrefs

Cf. A240584, A240585.

Sequence in context: A046002 A347695 A132257 * A132255 A142041 A190799

Adjacent sequences: A240580 A240581 A240582 * A240584 A240585 A240586

Keyword

nonn

Author

Michel Marcus, Apr 08 2014

Status

approved