A000924 -id:A000924 - cp's OEIS frontend

A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree.

0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 0, 3, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 2, 0, 2

Offset: 1

Jianing Song, Sep 25 2018

nonn

The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003643).

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Cf. A000924, A003643, A005117, A033197, A319659.

Real discriminant case: A317992.

PrimeNu[#*If[Mod[-#, 4]>1, 4, 1]] - 1& /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Aug 02 2019 *)

for(n=1, 200, if(issquarefree(n), print1(omega(n*if((-n)%4>1, 4, 1)) - 1, ", ")))

def A319662_list(len):
    L = []
    for n in (1..len):
        if is_squarefree(n):
            if (-n) % 4 > 1: n <<= 2
            L.append(sloane.A001221(n) - 1)
    return L
print(A319662_list(141)) # Peter Luschny, Oct 15 2018

a(n) = log_2(A003643(n)) = omega(A005117(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

Original entry on oeis.org

1, 1, 2, 1, 1, 8, 2, 10, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 2968, 15, 298, 16, 2, 5, 17, 1856, 1, 1, 9384, 97, 10, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 1084152, 117, 2, 746, 10, 88, 157, 126890, 1, 1, 1311, 56, 287

Offset: 1

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039955.

Subsequence of A077058 excluding terms for which A077425(n) is not squarefree. - Max Alekseyev, Dec 12 2012

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

Emmanuel Vantieghem, Table of n, a(n) for n=1..1000
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A053370-A053375.

2*NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 2]] & /@ Select[ Range[5, 309, 4], SquareFreeQ ]  (* Jean-François Alcover, Jul 09 2013 *)

forstep(n=5,1000,4, if(!issquarefree(n),next); print1( 2*polcoeff(lift(bnfinit(x^2-n).fu[1]),1), ", " )) /* Max Alekseyev */

A106030 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D=m if m=1 (mod 4), D=4*m otherwise and m>1 is the n-th squarefree number.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 1, 2, 4, 1, 2, 2, 2, 2, 2, 1, 4, 2, 2, 3, 4, 1, 2, 4, 1, 4, 2, 2, 2, 4, 1, 4, 2, 2, 2, 1, 2, 2, 4, 2, 2, 4, 2, 1, 2, 2, 4, 4, 3, 2, 2, 2, 4, 1, 4, 2, 2, 4, 1

Offset: 1

Steven Finch, May 05 2005

nonn

A104888 is the same except it is under the action of SL_2[Z].

			m = 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, ...
with corresponding discriminant
D = 8, 12, 5, 24, 28, 40, 44, 13, 56, 60, 17, ....

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Jens Jonasson, Classes of integral binary quadratic forms, Master's thesis (2001), Appendix B.

Cf. A104888.

A107996 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.

Original entry on oeis.org

5, 13, 21, 29, 45, 53, 61, 69, 77, 85, 93, 109, 117, 125, 133, 149, 157, 165, 173, 181, 205, 213, 221, 229, 237, 245, 253, 261, 277, 285, 293, 301, 309, 317, 341, 357, 365, 397, 413, 421, 429, 437, 445, 453, 461, 469, 477, 493, 501, 509, 517, 525, 533, 541

Offset: 1

Steven Finch, Jun 13 2005

nonn

From Wolfdieter Lang, Oct 30 2015: (Start)
These numbers m are the members of A079896 that have two conjugacy classes of proper solutions (and one of improper solutions) for the Pell equation x^2 - m*y^2 = +4. E.g., m = 5 has the proper positive fundamental solutions (3,1) and (7,3) obtained from (3,-1) (and the improper positive fundamental solution (18,8) = 2*(9,4) obtained from (2,0)).
For these numbers m one has therefore two conjugacy classes of improper solutions, and, in addition, the improper ambiguous class with member (4, 0) for the equation X^2 - m*Y^2 = +16.
Note that also even m may have solutions with both x and y odd, e.g., m = 12 with minimal positive solution (x, y) = (4, 1) for the +4 equation. The +-4 in the name means +4 or -4 (inclusive).
(End)

F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.
A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod. 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
N. Ishii, P. Kaplan and K. S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.
Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations.

Cf. A079896.

A107999 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Original entry on oeis.org

37, 101, 141, 189, 197, 269, 325, 333, 349, 373, 381, 389, 405, 485, 557, 573, 677, 701, 709, 757, 781, 813, 829, 877, 885, 901, 909, 925, 933, 973, 997, 1053, 1149, 1157, 1173, 1213, 1269, 1293, 1301, 1325, 1389, 1405, 1421, 1445, 1485, 1605, 1613, 1701, 1717

Offset: 1

Steven Finch, Jun 13 2005

nonn

C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966, section 256 VI, pp. 276-277.

A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod. 8), J. Reine Angew. Math. 53 (1857) 369-371.
S. R. Finch, Class number theory [Cached copy, with permission of the author]
N. Ishii, P. Kaplan and K. S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.

Cf. A048941, A048942, A107996, A108160.

More terms from Jinyuan Wang, Sep 08 2021

A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Original entry on oeis.org

37, 101, 141, 197, 269, 349, 373, 381, 389, 485, 557, 573, 677, 701, 709, 757, 781, 813, 829, 877, 885, 901, 933, 973, 997, 1149, 1157, 1173, 1213, 1293, 1301, 1389, 1405, 1605, 1613, 1717, 1757, 1765, 1861, 1885, 1893, 1901, 1909, 1949, 1973, 2069, 2077, 2093

Offset: 1

Steven Finch, Jun 13 2005

nonn

C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966, section 256 VI, pp. 276-277.

Florian Breuer and James Punch, Quadratic units and cubic fields, arXiv:2507.06579 [math.NT], 2025. See p. 1.
Arthur Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Noburo Ishii, Pierre Kaplan and Kenneth S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.

Cf. A048941, A048942, A107997, A107999.

More terms from Jinyuan Wang, Sep 08 2021

A175640 Decimal expansion of Product_{p = prime} (1 +(3p^2-1)/((p^2-1)p*(p+1)) ).

Original entry on oeis.org

2, 5, 9, 6, 5, 3, 6, 2, 9, 0, 4, 5, 0, 5, 4, 2, 0, 7, 3, 6, 3, 2, 7, 4, 0, 6, 5, 6, 6, 6, 9, 5, 1, 6, 1, 4, 2, 3, 7, 3, 9, 4, 6, 3, 0, 5, 2, 3, 4, 5, 0, 1, 4, 6, 2, 3, 6, 1, 5, 3, 6, 4, 9, 8, 1, 0, 6, 7, 5, 4, 8, 2, 4, 5, 7, 8, 7, 6, 0, 9, 3, 5, 2, 1, 9, 3, 7, 1, 2, 2, 2, 8, 7, 0, 2, 8, 6, 4, 3, 1, 4, 2, 8, 7, 4

Offset: 1

R. J. Mathar, Aug 01 2010

Named Barban's constant after the Soviet mathematician Mark Borisovich Barban (1935-1968). - Amiram Eldar, Mar 18 2021

			2.596536290450542073632740...

M. B. Barban, The large sieve method and its application to number theory, Russ. Math. Surv., Vol. 21, No. 1 (1966), pp. 49-103; MR 0199171.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 88.
Eric Weisstein's World of Mathematics, Barban's Constant.
Eric Weisstein's World of Mathematics, Prime Products
Wikipedia, Euler Product.

Cf. A000005, A000010.

read("transforms") : efact := 1+(3*p^2-1)/(p^2-1)/p/(p+1) ; Digits := 130 : tm := 380 : subs (p=1/x,1/efact) ; taylor(%,x=0,tm) : L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] : Le := EULERi(L) : x := 1.0 :
for i from 2 to nops(Le) do x := x/evalf(Zeta(i))^op(i,Le) ; x := evalf(x) ; print(x) ; end do:

digits = 50; $MaxExtraPrecision = 5 digits; s = Log[(1 + (3*p^2 - 1)/((p^2 - 1)*p*(p + 1)))] + O[p, Infinity]^(12 digits) // Normal; B = Exp[s /. Power[p, k_] -> PrimeZetaP[-k]]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Jul 24 2017 *)

prodeulerrat(1 +(3*p^2-1)/((p^2-1)*p*(p+1))) \\ Amiram Eldar, Mar 18 2021

Equals (29/18)*(61/48)*(397/360)*(1417/1344)*... inserting p = 2, 3, 5, 7, ... into the factor.

Equals Sum_{n>=1} A000005(n^2)*A000010(n)/n^3. - Richard R. Forberg, May 28 2023

More digits from Jean-François Alcover, Jul 24 2017

More digits from Vaclav Kotesovec, Jan 13 2021

A265643 a(n) = +-1 == ((p - 1)/2)! (mod p), where p is the n-th prime number == 3 (mod 4).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, -1

Offset: 1

Carlo Sanna, Dec 11 2015

sign

By Wilson's theorem, ((p - 1)/2)!^2 == (-1)^((p + 1)/2) (mod p) for each prime number p. Hence, if p == 3 (mod 4), then ((p - 1)/2)! == +-1 (mod p).
Michele Elia proved that a(n) = (-1)^((1 + h(-p)) / 2) for n > 1, where p is the n-th prime number == 3 (mod 4), and h(-p) is the class number of the quadratic field Q(sqrt(-p)).
Mordell (1961) proved the same result 52 years earlier in a 2-page note in the Monthly. - Jonathan Sondow, Apr 09 2017

			The second prime number == 3 (mod 4) is 7. Since ((7 - 1)/2)! = 3! = 6 == -1 (mod 7), it follows that a(2) = -1.

Robert Israel, Table of n, a(n) for n = 1..10000
Michele Elia, A note on the sequence ((p-1)/2)! mod p, International Mathematical Forum, 2013, Vol. 8, no. 37, pages 1813-1825.
L.J. Mordell, The congruence (p-1/2)! == +-1 (mod p), Amer. Math. Monthly, 68 (1961), 145-146.

Cf. A000924, A002145, A004055.

map(p -> if isprime(p) then mods(((p-1)/2)!, p) fi, [seq(i,i=3..10000, 4)]); # Robert Israel, Dec 11 2015

Function[p, Mod[((p-1)/2)!, p, -1]] /@ Select[Range[3, 2003, 4], PrimeQ] (* Jean-François Alcover, Feb 27 2016 *)

A053371 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of x for n == 2 mod 4.

Original entry on oeis.org

1, 5, 3, 15, 197, 5, 11, 35, 37, 13, 24335, 99, 63, 65, 251, 43, 53, 9, 10405, 2143295, 101, 4005, 21, 1025, 306917, 11, 57, 145925, 47, 143, 145, 21295, 7743, 1700902565, 13, 1451, 1601, 27, 7501, 52021, 195, 3141, 59535, 29

Offset: 0

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039956.

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A053370-A053375.

NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 1]] & /@ Select[Range[2, 210, 2], SquareFreeQ] (* Jean-François Alcover, Jul 09 2013 *)

A053372 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of x for n == 3 mod 4.

Original entry on oeis.org

2, 8, 10, 4, 170, 24, 1520, 6, 25, 3482, 48, 50, 89, 530, 48842, 3480, 80, 82, 28, 1574, 39, 227528, 962, 295, 1126, 120, 122, 4730624, 10610, 77563250, 12, 1728148040, 249, 1324, 64080026, 168, 4190210, 487, 1682, 8994000, 14

Offset: 0

N. J. A. Sloane, Jan 06 2000

Entries are indexed by values of n from A039957.

R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A053370-A053375.

NumberFieldFundamentalUnits[ Sqrt[#] ][[1, 2, 1]] & /@ Select[Range[3, 195, 4], SquareFreeQ ] (* Jean-François Alcover, Jul 09 2013 *)

cp's OEIS Frontend

A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree.

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A053373 Write fundamental unit for real quadratic field of discriminant n as x + y*omega; sequence gives values of y for n == 1 (mod 4).

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A106030 a(n) is the number of orbits under the action of GL_2[Z] on the primitive binary quadratic forms of discriminant D, where D=m if m=1 (mod 4), D=4*m otherwise and m>1 is the n-th squarefree number.

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A107996 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.

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A107999 Integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

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Extensions

A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

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Extensions

A175640 Decimal expansion of Product_{p = prime} (1 +(3*p^2-1)/((p^2-1)*p*(p+1)) ).

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Comments

Examples

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A265643 a(n) = +-1 == ((p - 1)/2)! (mod p), where p is the n-th prime number == 3 (mod 4).

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Maple

Mathematica

A175640 Decimal expansion of Product_{p = prime} (1 +(3p^2-1)/((p^2-1)p*(p+1)) ).