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A046927
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Number of ways to express 2n+1 as p+2q where p and q are primes.
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41
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0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, 4, 5, 4, 6, 4, 4, 7, 5, 3, 7, 3, 3, 7, 7, 5, 7, 4, 4, 8, 7, 5, 8, 4, 7, 8, 7, 4, 11, 5, 6, 9, 6, 5, 12, 6, 6, 10, 8, 6, 11, 7, 5, 11, 8, 6, 10, 6, 6, 13, 8, 5, 13, 6, 9, 12, 8, 6, 14, 8, 6, 11, 10, 9, 16, 5, 8, 13, 9, 9, 14, 7, 6, 14
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OFFSET
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0,5
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COMMENTS
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This is related to a conjecture of Lemoine (also sometimes called Levy's conjecture, although Levy was anticipated by Lemoine 69 years earlier). - Zhi-Wei Sun, Jun 10 2008
The conjecture states that any odd number greater than 5 can be written as p+2q where p and q are primes.
It can be conjectured that 1, 3, 5, 59 and 151 are the only odd integers n such that n + 2p and n + 2q both are composite for all primes p,q with n = p + 2q. (Following an observation from V. Shevelev, cf. link to SeqFan list.) - M. F. Hasler, Apr 10 2017
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REFERENCES
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L. E. Dickson, "History of the Theory of Numbers", Vol. I (Amer. Math. Soc., Chelsea Publ., 1999); see p. 424.
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LINKS
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E. Lemoine, L'intermédiaire des math., 1 (1894), p. 179; 3 (1896), p. 151.
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FORMULA
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For n >= 1, a(n) = Sum_{3<=p<=n+1, p prime} A((2*n + 1 - p)/2) + Sum_{2<=q<=(n+1)/2, q prime} B(2*n + 1 - 2*q) - A((n+1)/2)*B(n+1) - a(n-1) - ... - a(0), where A(n) = A000720(n), B(n) = A033270(n). - Vladimir Shevelev, Jul 12 2013
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MATHEMATICA
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a[n_] := (ways = 0; Do[p = 2k + 1; q = n-k; If[PrimeQ[p] && PrimeQ[q], ways++], {k, 1, n}]; ways); Table[a[n], {n, 0, 91}] (* Jean-François Alcover, Dec 05 2012 *)
Table[Count[FrobeniusSolve[{1, 2}, 2 n + 1], {__?PrimeQ}], {n, 0, 91}] (* Jan Mangaldan, Apr 08 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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