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%I #62 Jan 02 2023 12:30:46
%S 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,
%T 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,
%U 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
%N Number of ways to express 2n+1 as p+2q where p and q are primes.
%C 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
%C The conjecture states that any odd number greater than 5 can be written as p+2q where p and q are primes.
%C 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
%D L. E. Dickson, "History of the Theory of Numbers", Vol. I (Amer. Math. Soc., Chelsea Publ., 1999); see p. 424.
%H T. D. Noe, <a href="/A046927/b046927.txt">Table of n, a(n) for n = 0..10000</a>
%H L. Hodges, <a href="http://www.jstor.org/pss/2690477">A lesser-known Goldbach conjecture</a>, Math. Mag., 66 (1993), 45-47.
%H E. Lemoine, L'intermédiaire des math., 1 (1894), <a href="http://hdl.handle.net/2027/uc1.$b415067?urlappend=%3Bseq=197">p. 179</a>; 3 (1896), <a href="http://hdl.handle.net/2027/uc1.$b415069?urlappend=%3Bseq=157">p. 151</a>.
%H H. Levy, <a href="https://www.jstor.org/stable/3613447">On Goldbach's Conjecture</a>, Math. Gaz. 47 (1963), 274.
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/0901.3102">Binary additive problems: recursions for numbers of representations</a>, arXiv:0901.3102 [math.NT], 2009-2013.
%H V. Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-April/017447.html">Re: New sequence</a>, SeqFan list, April 2017.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LevysConjecture.html">Levy's Conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F 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
%t 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 *)
%t Table[Count[FrobeniusSolve[{1, 2}, 2 n + 1], {__?PrimeQ}], {n, 0, 91}] (* _Jan Mangaldan_, Apr 08 2013 *)
%o (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
%Y Cf. A194831 (records), A194830 (positions of records).
%K nonn
%O 0,5
%A _David W. Wilson_
%E Additional references from _Zhi-Wei Sun_, Jun 10 2008
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