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A249271
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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).
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0
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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
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OFFSET
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1,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants: Quadratic residues, pp. 96—98.
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LINKS
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Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation 35 (1980), pp. 1391-1417.
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FORMULA
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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.
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EXAMPLE
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3.147755148502400312516674955879769209272937748793398864...
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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