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A060651
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Smallest odd prime p such that Q(sqrt(-p)) has class number 2n+1.
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1
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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
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OFFSET
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0,1
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COMMENTS
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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)
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LINKS
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Table of n, a(n) for n=0..45.
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FORMULA
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a(n) = min(A002146(n), A002148(n)). - Jianing Song, Jul 20 2022
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MATHEMATICA
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<< 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 *)
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PROG
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(PARI) a(n) = forprime(p=3, oo, if ((p % 4) == 3, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022
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CROSSREFS
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Cf. A002146, A002148.
Sequence in context: A160022 A307530 A187094 * A146592 A107169 A297956
Adjacent sequences: A060648 A060649 A060650 * A060652 A060653 A060654
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v, Apr 17 2001
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EXTENSIONS
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Offset corrected by Michel Marcus, Jul 20 2022
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STATUS
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approved
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