A232927 -id:A232927 - cp's OEIS frontend

A249270 Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

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

Offset: 1

Jean-François Alcover, Oct 24 2014

The old definition was "Decimal expansion of the mean value over all positive integers of the least prime not dividing a given integer."

The integer parts of the sequence having this constant as starting value and thereafter x[n+1] = (frac(x[n])+1)*floor(x[n]), where floor and frac are integer and fractional part, are exactly the sequence of the prime numbers: see the Grime-Haran Numberphile video for details. - M. F. Hasler, Nov 28 2020

			2.9200509773161347120925629171120194680027278993214267...

Steven R. Finch, Meissel-Mertens constants: Quadratic residues, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 96—98.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
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. 171.
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73, ResearchGate link.
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.
Paul Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
I. A. Weinstein, Family of prime-representing constants: use of the ceiling function, arXiv:2101.00094 [math.GM], 2021.

Cf. A000040, A001223, A002110, A034386, A053669, A098990, A232927.

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

def sharp_primorial(n): return sloane.A002110(prime_pi(n));
@CachedFunction
def spv(n):
    b = 0
    for i in (0..n):
        b += 1 / sharp_primorial(i)
    return b
N(spv(300), digits=108) # Jani Melik, Jul 22 2015

Sum_{k >= 1} (p_k - 1)/(p_1 p_2 ... p_{k-1}), where p_k is the k-th prime number.
Sum_{k >= 0} 1/A034386(k). - Jani Melik, Jul 22 2015
From Amiram Eldar, Oct 29 2020: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053669(k).
Equals 2 + Sum_{n>=1} (prime(n+1)-prime(n))/prime(n)# = 2 + Sum_{n>=1} A001223(n)/A002110(n). (End)
prime(n+1) = floor(C*prime(n)# - prime(n)*floor(C*prime(n-1)# - 1)) with prime(1)=2 where C is this constant. - Davide Rotondo, Sep 15 2023

Definition revised by N. J. A. Sloane, Nov 29 2020

A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 3, 5, 2, 3, 2, 7, 5, 2, 5, 2, 7, 3, 3, 2, 3, 5, 13, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 5, 2, 3, 11, 2, 3, 11, 7, 7, 2, 7, 3, 3, 2, 7, 2, 3, 11, 2, 3, 2, 5, 5, 2, 5, 2, 11, 3, 3, 5, 2, 7, 11, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 7, 3, 11, 2, 3, 7, 7, 5, 2, 5, 2, 13, 3, 3, 2, 2, 3, 2, 3, 2, 5, 5, 11, 2, 7, 5, 3, 3, 5, 2, 3, 13, 5, 2, 3, 2, 17, 2, 2, 7, 3, 3, 2, 13, 2, 5, 2, 3, 5, 7, 5, 2, 5, 2, 11, 3, 2, 5, 2, 3, 7, 2, 3, 2, 17, 5, 7, 2, 7, 2, 5, 3, 3, 7, 2, 3, 7, 5, 2, 3

Offset: 2

Steven Finch, Dec 02 2013

nonn

From Jianing Song, Jan 30 2019: (Start)
a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.
a(n) is the least inert prime in the real quadratic field with discriminant D, D = A003658(n). (End)

			A003658(3) = 8, (8/3) = -1 and (8/2) = 0, so a(3) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Andrew Granville, R. A. Mollin and H. C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52:2 (2000), pp. 369-380.
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.
Enrique Treviño, The least inert prime in a real quadratic field, Mathematics of Computation 81:279 (2012), pp. 1777-1797. See also his PANTS 2010 talk.

Cf. A003658, A092419, A241482.

nMax = 200; A003658 = Select[Range[4nMax], NumberFieldDiscriminant[Sqrt[#]] == #&]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k]]]; a[n_] := f[A003658[[n]]]; Table[a[n], {n, 2, nMax}] (* Jean-François Alcover, Nov 05 2016 *)

lp(D)=forprime(p=2,,if(kronecker(D,p)<0,return(p)))
for(n=5,1e3,if(isfundamental(n),print1(lp(n)", "))) \\ Charles R Greathouse IV, Apr 23 2014

With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams proved, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the best known upper bound for the exponent is less than 0.16 when D is prime and 1/4 + epsilon (for any epsilon > 0) for general D. - Charles R Greathouse IV, Apr 23 2014 (corrected by Enrique Treviño, Mar 18 2022)

a(n) = A092419(A003658(n) - floor(sqrt(A003658(n)))), n >= 2. - Jianing Song, Jan 30 2019

Name simplified by Jianing Song, Jan 30 2019

A098990 Decimal expansion of Sum_{n>=1} prime(n)/(2^n).

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 07 2004

Relates the growth of the n-th prime function A000040(n) to the base-2 exponential of n.

			3.6746439660113287789956763090840294116777975887794373283122052201763...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.2.1, p. 96.

Thomas Bloom, Is Sum p_n/2^n irrational? (Here p_n is the nth prime), Erdős Problems.
Peter J. Cho and Henry H. Kim, The average of the smallest prime in a conjugacy class, International Mathematics Research Notices, Vol. 2020, No. 6 (2020), pp. 1718-1747, arXiv preprint, arXiv:1601.03012 [math.NT], 2016.
Paul Erdős, Remarks on number theory. I., Mat. Lapok, Vol. 12 (1961), pp. 10-17; Math. Rev. 26 #2410.
Steven R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
Paul Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202, alternative link.
Terence Tao, Erdős problem database, see no. 251.

Cf. A000040, A053760, A098882.

f:=N->sum(ithprime(n)/2^n,n=1..N); evalf[106](f(500)); evalf[106](f(1000));

RealDigits[Sum[Prime[i]/2^i,{i,1000}],10,120][[1]] (* Harvey P. Dale, Apr 10 2012 *)

suminf(k=1, prime(k)/2^k) \\ Michel Marcus, Jan 13 2016

Equals Sum_{n>=1} prime(n)/2^n.
Equals 2 plus the constant in A098882. - R. J. Mathar, Sep 02 2008
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053760(k). - Amiram Eldar, Oct 29 2020

A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

2, 3, 3, 5, 2, 7, 2, 11, 5, 13, 3, 2, 7, 3, 2, 5, 2, 3, 3, 11, 2, 2, 5, 7, 3, 2, 13, 5, 3, 2, 7, 3, 11, 2, 11, 2, 7, 11, 7, 2, 3, 2, 5, 3, 2, 5, 3, 3, 5, 2, 11, 2, 13, 5, 5, 2, 5, 3, 2, 7, 2, 3, 2, 2, 5, 13, 2, 3, 2, 5, 17, 3, 2, 7, 3, 3, 5, 2, 13, 2, 7, 5, 19, 2, 3, 11, 3, 2, 5, 2, 3, 3, 7, 2, 5, 2, 5, 11, 5, 3, 2, 5, 3, 2, 11, 2, 3, 7, 2, 2, 11, 7, 3, 2, 5, 3, 2, 5, 3, 3, 2, 11, 2, 19, 5, 5, 2, 3, 2, 17, 3, 2, 7, 2, 3, 3, 13, 2, 5, 2, 5, 11, 7, 3, 2, 7, 3, 13, 2, 3, 5, 2, 2

Offset: 1

Steven Finch, Dec 02 2013

nonn

From Jianing Song, Feb 14 2019: (Start)
a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.
a(n) is the least inert prime in the imaginary quadratic field with discriminant D, D = -A003657(n). (End)

			A003657(4) = 8, (-8/5) = -1, (-8/3) = 1 and (-8/2) = 0, so a(4) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
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. A003657, A306220.

nMax = 200; FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]]; discrims = Select[-Range[4 nMax], FundamentalDiscriminantQ]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k] ] ]; a[n_] := f[discrims[[n]]]; Table[a[n], {n, 1, nMax}] (* Jean-François Alcover, Nov 05 2016, after Robert G. Wilson v *)

a(n) = A306220(A003657(n)). - Jianing Song, Feb 14 2019

Name simplified by Jianing Song, Feb 14 2019

A232929 For each complex nonprincipal Dirichlet character chi modulo n, let f(chi) be the least positive integer k for which chi(k) is not in the set {0,1}. Then a(n) is the sum of f(chi) over all such chi.

Original entry on oeis.org

2, 3, 6, 5, 11, 11, 10, 9, 18, 17, 22, 15, 19, 23, 31, 25, 34, 29, 25, 31, 45, 47, 38, 39, 34, 35, 54, 53, 63, 47, 41, 45, 47, 57, 70, 51, 51, 61, 79, 61, 84, 61, 51, 65, 93, 87, 83, 57, 71, 75, 102, 85, 79, 81, 73, 81, 114, 119, 118, 87, 85, 95, 97, 97, 130, 95, 89, 85, 143, 127, 151, 107, 83, 109, 119, 125, 155, 125, 106, 125, 162, 135, 133, 123, 113, 125, 181, 165, 147, 131, 139, 137, 147, 167, 193, 123, 121, 125, 198, 157, 203, 161, 123, 153, 210, 177, 216, 121, 151, 153, 225, 183, 179, 169, 159, 179, 201, 255

Offset: 3

Steven Finch, Dec 02 2013

nonn

			a(6) = 5 since there is one nonprincipal Dirichlet character mod 6, namely A134667, whose fifth term is -1.

S. 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 Erdos, J. London Math. Soc. 87 (2013) 22-42.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000010.

A232930 For each complex primitive Dirichlet character chi modulo n, let f(chi) be the least positive integer k for which chi(k) is not in the set {0,1}. Then a(n) is the sum of f(chi) over all such chi.

Original entry on oeis.org

2, 3, 6, 0, 11, 8, 8, 0, 18, 5, 22, 0, 11, 12, 31, 0, 34, 17, 10, 0, 45, 20, 32, 0, 24, 17, 54, 0, 63, 24, 21, 0, 30, 20, 70, 0, 27, 22, 79, 0, 84, 27, 24, 0, 93, 20, 72, 0, 36, 33, 102, 0, 55, 38, 37, 0, 114, 27, 118, 0, 52, 48, 69, 0, 130, 47, 42, 0, 143, 40, 151, 0, 32, 55, 90, 0, 155, 52, 72, 0, 162, 33, 96, 0, 57, 56, 181, 0, 114, 63, 58, 0, 107, 40, 193, 0, 72, 48, 198, 0, 203, 78, 39, 0, 210, 60, 216, 0, 79, 60, 225, 0, 126, 85, 100, 0, 159, 46

Offset: 3

Steven Finch, Dec 02 2013

nonn

			a(6)=0 since there are no primitive Dirichlet characters mod 6.

S. 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 Erdos, J. London Math. Soc. 87 (2013) 22-42.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A007431.

A232928 a(n) is the smallest q such that the primes<=q generate the multiplicative group modulo n.

Original entry on oeis.org

2, 3, 2, 5, 3, 5, 2, 3, 2, 7, 2, 3, 7, 5, 3, 5, 2, 11, 5, 7, 5, 13, 2, 5, 2, 5, 2, 11, 3, 5, 5, 3, 3, 7, 2, 3, 7, 11, 3, 11, 3, 7, 7, 5, 5, 13, 3, 3, 5, 5, 2, 5, 3, 11, 5, 3, 2, 13, 2, 3, 5, 5, 3, 7, 2, 5, 5, 19, 7, 13, 5, 5, 7, 7, 3, 7, 3, 11, 2, 5, 2, 13, 3, 3, 5, 7, 3, 11, 3, 5, 11, 5, 7, 13, 5, 3, 5, 11, 2, 7, 3, 11, 13, 3, 2, 7, 3, 7, 11, 11, 3, 13, 7, 7, 7, 11, 11, 17

Offset: 3

Steven Finch, Dec 02 2013

nonn

E. Bach and L. Huelsbergen, Statistical evidence for small generating sets, Math. Comp. 61 (1993) 69-82.
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdos, J. Number Theory 132 (2012) 1185-1202.

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

A249270 Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

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A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

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A098990 Decimal expansion of Sum_{n>=1} prime(n)/(2^n).

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A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

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A232929 For each complex nonprincipal Dirichlet character chi modulo n, let f(chi) be the least positive integer k for which chi(k) is not in the set {0,1}. Then a(n) is the sum of f(chi) over all such chi.

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A232930 For each complex primitive Dirichlet character chi modulo n, let f(chi) be the least positive integer k for which chi(k) is not in the set {0,1}. Then a(n) is the sum of f(chi) over all such chi.

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A232928 a(n) is the smallest q such that the primes<=q generate the multiplicative group modulo n.

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

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