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 A046927 Number of ways to express 2n+1 as p+2q where p and q are primes. 37
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 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 REFERENCES L. E. Dickson, "History of the Theory of Numbers", Vol. I (Amer. Math. Soc., Chelsea Publ., 1999); see p. 424. LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47. E. Lemoine, L'intermédiaire des math., 1 (1894), p. 179; 3 (1896), p. 151. H. Levy, On Goldbach's Conjecture, Math. Gaz. 47 (1963), 274. Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013. V. Shevelev, Re: New sequence, SeqFan list, April 2017. Eric Weisstein's World of Mathematics, Levy's Conjecture FORMULA 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 MATHEMATICA 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 *) PROG (PARI) a(n)=my(s); n=2*n+1; forprime(p=2, n\2, s+=isprime(n-2*p)); s \\ Charles R Greathouse IV, Jul 17 2013 CROSSREFS Cf. A194831 (records), A194830 (positions of records). Sequence in context: A061389 A138011 A036555 * A084718 A154851 A281854 Adjacent sequences:  A046924 A046925 A046926 * A046928 A046929 A046930 KEYWORD nonn AUTHOR EXTENSIONS Additional references from Zhi-Wei Sun, Jun 10 2008 STATUS approved

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)