A249271 Decimal expansion of the mean value over all positive integers of a function giving the least quadratic nonresidue modulo a given odd integer (this function is precisely defined in A053761).

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

Offset: 1

Author

Jean-François Alcover, Oct 24 2014

Keywords

• nonn
• cons

Examples

			3.147755148502400312516674955879769209272937748793398864...
		

References

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

Links

• Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation 35 (1980), pp. 1391-1417.

Crossrefs

Cf. A053760, A053761, A098990.

Programs

Mathematica

digits = 104; Clear[s]; s[m_] := s[m] = 1 + Sum[(Prime[j] + 1)*2^(-j + 1)* Product[1 - 1/Prime[i], {i, 1, j - 1}] // N[#, digits + 100]&, {j, 2, m}] ; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], Print[m]; m = 2*m]; RealDigits[s[m], 10, digits] // First

PARI

do(lim)=my(p=2,pr=1.,s=1); forprime(q=3,lim, pr*=(1-1/p)/2; s+=(q+1)*pr; p=q); s \\ Charles R Greathouse IV, Dec 20 2017

Formula

1 + sum_{j=2..m} (p_j + 1)*2^(-j+1)*prod_{i=1..j-1} (1 - 1/p_i), where p_j is the j-th prime number.