A046927 Number of ways to express 2n+1 as p+2q where p and q are primes.

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

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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 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

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
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A194831 (records), A194830 (positions of records).

Programs

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 *)

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

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

Extensions

Additional references from Zhi-Wei Sun, Jun 10 2008