A249272 Decimal expansion of a constant associated with fundamental discriminants and Dirichlet characters.
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
Examples
4.9809473396149341507913253258807752812377326965852...
Links
- 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.
Programs
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Mathematica
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 -
PARI
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
Formula
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.
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
Examples
2.5350541803604388301655300071859083508611780138537...
Links
- 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.
Programs
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Mathematica
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
Formula
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.
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
Examples
2.1514351056861465486242810050965840532633...
Links
- 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.
Programs
-
Mathematica
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
Formula
sum_{q} q^4/((q+1)^2 (q-1)) prod_{p