A232931 -id:A232931 - cp's OEIS frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197

Offset: 1

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)
Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Fundamental Discriminant.
Eric Weisstein's World of Mathematics, Class Number.

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)

v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v

list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022

def is_fundamental(d):
    r = d % 4
    if r > 1 : return False
    if r == 1: return (d != 1) and is_squarefree(d)
    q = d // 4
    return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).

a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - Charles R Greathouse IV, Jan 21 2022

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

7, 5, 2, 3, 3, 2, 5, 3, 2, 5, 2, 3, 2, 7, 11, 2, 5, 7, 2, 3, 3, 17, 3, 2, 2, 3, 5, 2, 13, 5, 2, 2, 3, 3, 2, 7, 3, 2, 11, 11, 2, 3, 7, 5, 5, 2, 19, 2, 3, 3, 2, 41, 3, 2, 13, 3, 2, 5, 7, 2, 7, 2, 3, 5, 3, 2, 5, 2, 3, 11, 2, 31, 13, 2, 5, 2, 3, 3, 2, 5, 3, 2, 5, 23, 2, 5, 17, 7, 2, 5, 7, 2, 3, 3

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that decomposes in the imaginary quadratic field with discriminant D, D = -A003657(n).
For most n, a(n) is relatively small. There are only 472 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)).
Also a(n) is the smallest prime p such that the imaginary quadratic field with discriminant D = -A003657(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

			Let K = Q[sqrt(-177)] with D = -708 = -A003657(218), we have: 2 and 3 divides 708, (-708/5) = (-708/7) = ... = (-708/29) = -1 and (-708/31) = +1, so 2 and 3 ramify in K, 5, 7, ..., 29 remain inert in K and 31 decomposes in K, so a(218) = 31.

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, this sequence (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))

A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 11, 7, 11, 3, 2, 5, 5, 3, 5, 2, 3, 3, 2, 5, 7, 5, 2, 7, 3, 2, 5, 2, 3, 13, 3, 3, 2, 7, 7, 2, 5, 5, 2, 3, 2, 7, 3, 2, 3, 3, 2, 13, 5, 2, 5, 11, 5, 3, 2, 7, 11, 3, 13, 2, 3, 3, 2, 11, 2, 7, 2, 5, 3, 2, 11, 2, 3, 5, 3, 3, 2, 5, 13, 2, 13, 2, 3, 2, 5, 2, 3, 5, 2

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).
For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).
Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

			Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

A306541 The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 5, 2, 2, 3, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 7, 2, 2, 2, 3, 2, 3, 2, 2, 7, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 3, 2, 2, 13, 2, 3, 2, 2, 2, 2, 7, 2, 2, 3, 2, 3, 2, 2, 5, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that either decomposes or ramifies in the real quadratic field with discriminant D, D = A003658(n).

For most n, a(n) is relatively small. There are only 86 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).

			Let K = Q[sqrt(293)] with D = 293 = A003658(90), we have: (293/2) = (293/3) = ... = (293/13) = -1 and (293/17) = +1, so 2, 3, 5, 7, 11 and 13 remain inert in K and 17 decomposes in K, so a(90) = 17.

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); this sequence, A306542 (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

3, 2, 2, 2, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 11, 2, 3, 2, 2, 2, 3, 17, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 2, 41, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 3, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 2, 2, 7

Offset: 1

Jianing Song, Feb 22 2019

nonn

a(n) is the least prime that either decomposes or ramifies in the imaginary quadratic field with discriminant D, D = -A003657(n).
The quadratic field with discriminant D = -A003657(n) has class number 1 if and only if a(n) >= (1/4)*A003657(n). If the quadratic field with discriminant D = -A003657(n) has class number 3 then a(n)^2 < (1/4)*A003657(n) < a(n)^3.
For most n, a(n) is relatively small. There are only 86 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)). In fact, if we ignore the first term, the only terms among the first 3043 ones that seem unusually large are a(15) = 11 (with A003657(15) = 43), a(22) = 17 (with A003657(22) = 67), a(52) = 41 (with A003657(52) = 163), a(1147) = 23 (with A003657(1147) = 3763), a(2677) = 23 (with A003657(2677) = 8803) and a(2758) = 23 (with A003657(2758) = 9067).

			Let K = Q[sqrt(-3763)] with D = -3763 = -A003657(1147), we have: (-3763/2) = (-3763/3) = ... = (-3763/19) = -1 and (-3763/23) = +1, so 2, 3, 5, 7, 11, 13, 17 and 19 remain inert in K and 23 decomposes in K, so a(1147) = 23.

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); A306541, this sequence (the least prime that decomposes or ramifies).

b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))

A241482 Least fundamental discriminant D > 1 such that the first n primes p have (D/p) >= 0.

Original entry on oeis.org

8, 12, 24, 60, 60, 364, 984, 1596, 1596, 1596, 3705, 58444, 84396, 164620, 172236, 369105, 369105, 731676, 731676, 3442296, 3442296, 32169916, 32169916, 47973864, 47973864, 47973864, 313114620, 313114620, 313114620, 313114620, 13461106065, 27765196680, 40527839121, 55213498824, 55213498824, 381031123720

Offset: 1

Charles R Greathouse IV, Apr 23 2014

nonn

By the Chinese Remainder Theorem and Prime Number Theorem in arithmetic progressions, this sequence is infinite.

a(n) is the least fundamental discriminant D > 1 such that the first n primes either decompose or ramify in the real quadratic field with discriminant D. See A306218 for the imaginary quadratic field case. - Jianing Song, Feb 14 2019

			(364/2) = 0, (364/3) = 1, (364/5) = 1, (364/7) = 0, (364/11) = 1, (364/13) = 0, so 3, 5, 11 decompose in Q[sqrt(91)] and 2, 7, 13 ramify in Q[sqrt(-231)]. For other fundamental discriminants 1 < D < 364, at least one of 2, 3, 5, 7, 11, 13 is inert in the imaginary quadratic field with discriminant D, so a(6) = 364. - _Jianing Song_, Feb 14 2019

Cf. A003658, A232931, A306218 (the imaginary quadratic field case).

A002189 and A094847 are similar sequences.

a(n) = my(i=2); while(!isfundamental(i)||sum(j=1, n, kronecker(i,prime(j))==-1)!=0, i++); i \\ Jianing Song, Feb 14 2019

a(n) > prime(n)^(4*sqrt(e) + o(1)). - Charles R Greathouse IV, Apr 23 2014

a(n) = A003658(k), where k is the smallest number such that A232931(k) >= prime(n+1). - Jianing Song, Feb 15 2019

a(36) from Charles R Greathouse IV, Apr 24 2014

A249272 Decimal expansion of a constant associated with fundamental discriminants and Dirichlet characters.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			4.9809473396149341507913253258807752812377326965852...

Peter J. Cho, Henry H. Kim, The average of the smallest prime in a conjugacy class, arXiv:1601.03012 [math.NT], 2016.
Steven R. Finch, Average least nonresidues, December 4, 2013. [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. 250.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.

Cf. A232929, A232930, A232931, A232932.

digits = 103; Clear[s, P]; P[j_] := P[j] = Product[(Prime[k] + 2)/(2*(Prime[k] + 1)), {k, 1, j - 1}] // N[#, digits + 100]&; s[m_] := s[m] = Sum[Prime[j]^2/(2*(Prime[j] + 1))*P[j], {j, 1, m}]; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

suminf(k=1, prime(k)^2/(2*(prime(k)+1))*prod(i=1, k-1, (prime(i)+2)/(2*(prime(i)+1)))); \\ Michel Marcus, Apr 15 2017

sum_{q} q^2/(2(q+1)) prod_{p

A249273 Decimal expansion of a constant associated with the set of all complex nonprincipal Dirichlet characters.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.5350541803604388301655300071859083508611780138537...

Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
G. Martin and P. Pollack, The average least character non-residue and further variations on a theme of Erdős, J. London Math. Soc. 87 (2013) 22-42.

Cf. A232929, A232930, A232931, A232932.

digits = 103; Clear[s]; s[m_] := s[m] = Sum[Prime[k]^2/Product[Prime[j] + 1, {j, 1, k}] , {k, 1, m}] // N[#, digits + 100]&; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

sum_{k >= 1} p_k^2/((p_1 + 1)(p_2 + 1)...(p_k + 1)), where p_k is the k-th prime number.

A249274 Decimal expansion of a constant associated with the set of all complex primitive Dirichlet characters.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 24 2014

			2.1514351056861465486242810050965840532633...

Steven R. Finch, Average least nonresidues, December 4, 2013. [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. 251.
G. Martin and P. Pollack, The average least character non-residue and further variations on a theme of Erdős, J. London Math. Soc. 87 (2013) 22-42.

Cf. A232929, A232930, A232931, A232932.

digits = 101; Clear[s, P]; P[j_] := P[j] = Product[(Prime[k]^2 - Prime[k] - 1)/((Prime[k] + 1)^2*(Prime[k] - 1)), {k, 1, j - 1}] // N[#, digits + 100]&; s[m_] := s[m] = Sum[Prime[j]^4/((Prime[j] + 1)^2*(Prime[j] - 1))*P[j], {j, 1, m}]; s[10]; s[m = 20]; While[ RealDigits[s[m]] != RealDigits[s[m/2]], Print[m, " ", N[s[m]]]; m = 2*m]; RealDigits[s[m], 10, digits] // First

sum_{q} q^4/((q+1)^2 (q-1)) prod_{p

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cp's OEIS Frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

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A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

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A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

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A306541 The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

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A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

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A241482 Least fundamental discriminant D > 1 such that the first n primes p have (D/p) >= 0.

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A249272 Decimal expansion of a constant associated with fundamental discriminants and Dirichlet characters.

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A249273 Decimal expansion of a constant associated with the set of all complex nonprincipal Dirichlet characters.

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