

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
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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).  ZhiWei 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 lesserknown Goldbach conjecture, Math. Mag., 66 (1993), 4547.
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], 20092013.
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


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(n1)  ...  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 = nk; If[PrimeQ[p] && PrimeQ[q], ways++], {k, 1, n}]; ways); Table[a[n], {n, 0, 91}] (* JeanFrançois 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(n2*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

David W. Wilson


EXTENSIONS

Additional comments and references from ZhiWei Sun, Jun 10 2008


STATUS

approved



