A002148 -id:A002148 - cp's OEIS frontend

A355877 Smallest prime p == 5 (mod 8) such that Q(sqrt(p)) has class number 2n+1.

5, 229, 1093, 2029, 7573, 12589, 8101, 13693, 54541, 18229, 75629, 91813, 59053, 65029, 72901, 146077, 127453, 199813, 169909, 209581, 439573, 189229, 197341, 324901, 378229, 596293, 430861, 352837, 712981, 1137229, 700573, 245029, 574261, 770533, 860701, 1432813, 1821877, 1092829

Offset: 0

Jianing Song, Jul 20 2022

nonn

			p = 229 is the smallest prime congruent to 5 modulo 8 such that Q(sqrt(p)) has class number 3, so a(1) = 229.

Cf. A355878.

Similar sequences: A355876 (p == 1 (mod 8)), A002148 (p == 3 (mod 8)), A002146 (p == 7 (mod 8)).

a(n) = forprime(p=2, oo, if(p%8==5 && qfbclassno(p)==2*n+1, return(p)))

A355878 Smallest p == 1 (mod 4) such that Q(sqrt(p)) has class number 2n+1.

Original entry on oeis.org

5, 229, 401, 577, 1129, 1297, 8101, 11321, 11257, 18229, 7057, 23593, 27689, 8761, 56857, 146077, 63361, 25601, 24337, 55441, 439573, 14401, 32401, 78401, 70969, 69697, 376897, 106537, 41617, 160001, 193601, 57601, 197137, 367721, 414433, 1432813, 444089, 331777

Offset: 0

Jianing Song, Jul 20 2022

nonn

Also smallest odd prime p such that Q(sqrt(p)) has narrow class number (also called form class number) 2n+1.

Conjecture: a(n) > A002148(n) for all n.

			p = 229 is the smallest odd prime such that Q(sqrt(p)) has class number 3, so a(1) = 229.

Cf. A355876, A355877.

a(n) = forprime(p=2, oo, if(p%4==1 && qfbclassno(p)==2*n+1, return(p)))

a(n) = min(A355876(n),A355877(n)).

A060651 Smallest odd prime p such that Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

3, 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

Offset: 0

Robert G. Wilson v, Apr 17 2001

nonn

Note that all such primes are congruent to 3 modulo 4.
Conjecture: a(n) = A002146(n) for all n >= 1. That is to say, A002148(n) > A002146(n) for all n >= 1. - Jianing Song, Jul 20 2022
From Jianing Song, Sep 16 2022: (Start)
Note that an imaginary quadratic field has an odd class number if and only if it is of the form Q(sqrt(-1)), Q(sqrt(-2)), or Q(sqrt(-p)) for primes p == 3 (mod 4).
It seems that for most n, the class group of Q(sqrt(-a(n))) is the cyclic group of order 2*n+1. But this is not always true. The smallest prime p such that Q(sqrt(-p)) has class number 243 is p = 29399, and the class group of Q(sqrt(-29399)) is C_3 X C_81 rather than C_243. Also, the smallest prime p such that Q(sqrt(-p)) has class number 637 is p = 149519, and the class group of Q(sqrt(-149519)) is C_7 X C_91 rather than C_637. (End)

Cf. A002146, A002148.

<< NumberTheory`NumberTheoryFunctions`
a = Table[0, {101}]; Do[ c = ClassNumber[ -Prime[n] ]; If[ c < 102 && a[ [c] ] == 0, a[ [c] ] = Prime[n] ], {n, 2, 4000} ]; Table[ a[ [n] ], {n, 1, 101} ]
a = Table[0, {101}]; Do[c = NumberFieldClassNumber[Sqrt[-Prime[n]]]; If[c < 102 && a[[c]] == 0, a[[c]] = Prime[n]], {n, 2, 4000}]; Select[ Table[a[[n]], {n, 1, 101}], Mod[#, 4] == 3 &] (* Jean-François Alcover, Jul 20 2022 *)

a(n) = forprime(p=3, oo, if ((p % 4) == 3, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022

a(n) = min(A002146(n), A002148(n)). - Jianing Song, Jul 20 2022

Offset corrected by Michel Marcus, Jul 20 2022

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A355877 Smallest prime p == 5 (mod 8) such that Q(sqrt(p)) has class number 2n+1.

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A355878 Smallest p == 1 (mod 4) such that Q(sqrt(p)) has class number 2n+1.

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A060651 Smallest odd prime p such that Q(sqrt(-p)) has class number 2n+1.

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