A002143 Class numbers h(-p) where p runs through the primes p == 3 (mod 4).
1, 1, 1, 1, 3, 3, 1, 5, 3, 1, 7, 5, 3, 5, 3, 5, 5, 3, 7, 1, 11, 5, 13, 9, 3, 7, 5, 15, 7, 13, 11, 3, 3, 19, 3, 5, 19, 9, 3, 17, 9, 21, 15, 5, 7, 7, 25, 7, 9, 3, 21, 5, 3, 9, 5, 7, 25, 13, 5, 13, 3, 23, 11, 5, 5, 31, 13, 5, 21, 15, 5, 7, 9, 7, 33, 7, 21, 3, 29, 3, 31, 19, 5, 11, 15, 27, 17, 13
Offset: 1
Keywords
Examples
E.g., a(4) = 1 is the class number of -19, the 4th prime == 3 mod 4. a(5) = -(1/23)*sum(n=1..22, n*(n|23)) = -(1/23)*(1 + 2 + 3 + 4 - 5 + 6 - 7 + 8 + 9 - 10 - 11 + 12 + 13 - 14 - 15 + 16 - 17 + 18 - 19 - 20 - 21 - 22) = 69/23 = 3. - _Jonathan Sondow_, Oct 27 2011
References
- H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.
- 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
- T. D. Noe, Table of n, a(n) for n=1..10000
- Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.
- Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
- E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]
- E. T. Ordman, Tables of the class number for negative prime discriminants, Math. Comp., 23 (1969), 458.
- A. Pacetti and F. Rodriguez Villegas, Computing weight two modular forms of level p^2, Math. Comp. 74 (2004), 1545-1557. See Table 1.
- N. Snyder, Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms. [Background information on class numbers, link sent by V. S. Miller, Nov 22 2009]
- Wikipedia, Class numbers of quadratic fields
Programs
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Mathematica
Cases[ Table[ With[ {p = Prime[n]}, If[ Mod[p, 4] == 3, -(1/p)*Sum[ a*JacobiSymbol[a, p], {a, 1, p - 1}]]], {n, 1, 100}], Integer] (* _Jonathan Sondow, Oct 27 2011 *) p = Prime[n]; If[Mod[p, 4] == 3, NumberFieldClassNumber[Sqrt[-p]]] (* Jonathan Sondow, Feb 24 2012 *) -
PARI
forprime(p=3, 1e3, if(p%4==3, print1(qfbclassno(-p)", "))) \\ Charles R Greathouse IV, May 08 2011
Formula
h(-p) = 1 + 2*sum(0 <= n <= (1/2)*sqrt(p/3)-1, d(n^2+n+(p+1)/4, [2*n+1, sqrt(n^2+n+(p+1)/4)])) for prime p=3 mod 4, p>3. d(n, [a, b])=card{d: d|n and a
h(-p) = -(1/p)*sum(n=1..p-1, n*(n|p)) if p > 3, where (n|p) = +/- 1 is the Legendre symbol. - Jonathan Sondow, Oct 27 2011
h(-p) = (1/3)*sum(n=1..(p-1)/2, (n|p)) or sum(n=1..(p-1)/2, (n|p)) according as p == 3 or 7 (mod 8). - Jonathan Sondow, Feb 27 2012
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 19 2002
A002147 Largest prime == 7 (mod 8) with class number 2n+1.
7, 31, 127, 487, 1423, 1303, 2143, 2647, 4447, 5527, 5647, 6703, 5503, 11383, 8863, 13687, 13183, 12007, 22807, 18127, 21487, 22303, 29863, 25303, 27127
Offset: 0
Keywords
Comments
Apr 14 2008: David Broadhurst says: I computed class numbers for prime discriminants with |D| < 10^9, but stopped when the first case with |D| > 5*10^8 was observed. That factor of 2 seems to me to be a reasonable margin of error, when you look at the pattern of what is included.
References
- 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, Table of n, a(n) for n = 0..2246 (conjectural; see comment).
- 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.
Crossrefs
Cf. A002146.
A355876 Smallest prime p == 1 (mod 8) such that Q(sqrt(p)) has class number 2n+1.
17, 257, 401, 577, 1129, 1297, 13033, 11321, 11257, 38569, 7057, 23593, 27689, 8761, 56857, 284561, 63361, 25601, 24337, 55441, 458929, 14401, 32401, 78401, 70969, 69697, 376897, 106537, 41617, 160001, 193601, 57601, 197137, 367721, 414433, 1506473, 444089, 331777, 156817
Offset: 0
Keywords
Comments
Examples
p = 257 is the smallest prime congruent to 1 modulo 8 such that Q(sqrt(p)) has class number 3, so a(1) = 257.
Links
Crossrefs
Programs
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PARI
a(n) = forprime(p=2, oo, if(p%8==1 && qfbclassno(p)==2*n+1, return(p)))
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
Keywords
Examples
p = 229 is the smallest prime congruent to 5 modulo 8 such that Q(sqrt(p)) has class number 3, so a(1) = 229.
Links
Crossrefs
Programs
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PARI
a(n) = forprime(p=2, oo, if(p%8==5 && qfbclassno(p)==2*n+1, return(p)))
A355879 Class number of Q(sqrt((-1)^((p-1)/2)*p)), where p = prime(n).
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 7, 1, 5, 3, 1, 1, 1, 5, 3, 1, 1, 5, 5, 1, 3, 1, 7, 1, 1, 11, 1, 5, 1, 13, 1, 1, 9, 3, 7, 5, 3, 1, 15, 1, 7, 3, 13, 1, 11, 1, 1, 3, 1, 3, 19, 1, 1, 3, 1, 5, 1, 1, 19, 9, 1, 3, 17, 1, 1, 5, 1, 9, 1, 21, 1, 15, 5, 1, 1, 1, 7
Offset: 1
Keywords
Comments
For n > 1, class number of the unique quadratic field with discriminant +-p, p = prime(n).
a(1) corresponds to Q(sqrt(2*i)) = Q(1+i) = Q(i).
All terms are odd.
Examples
prime(9) = 23, Q(sqrt(-23)) has class number 3, so a(9) = 3. prime(15) = 47, Q(sqrt(-47)) has class number 5, so a(15) = 5. prime(20) = 71, Q(sqrt(-71)) has class number 7, so a(20) = 7. prime(50) = 229, Q(sqrt(229)) has class number 3, so a(50) = 3.
Links
Programs
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PARI
a(n) = if(n==1, 1, my(p=prime(n)); qfbclassno(if(p%4==1, p, -p)))
A060651 Smallest odd prime p such that Q(sqrt(-p)) has class number 2n+1.
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
Keywords
Comments
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)
Programs
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Mathematica
<< 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 *) -
PARI
a(n) = forprime(p=3, oo, if ((p % 4) == 3, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022
Formula
Extensions
Offset corrected by Michel Marcus, Jul 20 2022
Comments