A002146 - OEIS

A002146

Smallest prime == 7 (mod 8) where Q(sqrt(-p)) has class number 2n+1.
(Formerly M4377 N1841)

7, 23, 47, 71, 199, 167, 191, 239, 383, 311, 431, 647, 479, 983, 887, 719, 839, 1031, 1487, 1439, 1151, 1847, 1319, 3023, 1511, 1559, 2711, 4463, 2591, 2399, 3863, 2351, 3527, 3719, 3119, 5471, 2999, 4703, 6263, 4391, 3671, 3911, 4079, 5279, 6311, 4679, 5351, 6959

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OFFSET

0,1

COMMENTS

Conjecture: a(n) < A002148(n) for all n >= 1. - Jianing Song, Jul 20 2022

REFERENCES

Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989, pp. 224-241.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

David Broadhurst and T. D. Noe, Table of n, a(n) for n = 0..28603

D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

MATHEMATICA

With[{p = Select[Prime[Range[1000]], Mod[#, 8] == 7 &]}, c = NumberFieldClassNumber[Sqrt[-p]]; p[[TakeWhile[Flatten[FirstPosition[c, #] & /@ Range[1, Max[c], 2]], NumberQ]]]] (* Amiram Eldar, Mar 16 2026 *)

PROG

(PARI) a(n) = forprime(p=2, oo, if ((p % 8) == 7, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022

CROSSREFS

Cf. A002147, A002148, A060651, A002143 (class numbers).

Sequence in context: A185007 A308732 A139035 * A336092 A184882 A073577

Adjacent sequences: A002143 A002144 A002145 * A002147 A002148 A002149

KEYWORD

nonn

AUTHOR

N. J. A. Sloane and Mira Bernstein

STATUS

approved