A194831 -id:A194831 - cp's OEIS frontend

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

David W. Wilson

nonn

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

L. E. Dickson, "History of the Theory of Numbers", Vol. I (Amer. Math. Soc., Chelsea Publ., 1999); see p. 424.

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

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

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 *)

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

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

Additional references from Zhi-Wei Sun, Jun 10 2008

A194830 Odd numbers for which the number of ways to express them as 2*p+q, p,q primes, is a new record.

Original entry on oeis.org

7, 9, 17, 33, 45, 51, 75, 93, 105, 135, 153, 165, 225, 273, 285, 315, 405, 465, 495, 525, 735, 765, 945, 1155, 1365, 1785, 1995, 2145, 2415, 2625, 3045, 3255, 3465, 3885, 4095, 4305, 4725, 4935, 5145, 5355, 5565, 5775, 6405, 6825, 7665, 8085, 8925, 9555

Offset: 1

Hugo Pfoertner, Sep 12 2011

nonn

2*(positions of records in A046927)+1. The corresponding new record number of representations is given in A194831.

			a(3)=17, because it can be represented in A194831(3)=4 different ways (17=2*2+13=2*3+11=2*5+7=2*7+3), whereas all smaller odd numbers can only be represented in a smaller number of ways.

See A046927.

Hugo Pfoertner, Table of n, a(n) for n = 1..322
Eric Weisstein's World of Mathematics, Levy's Conjecture

Cf. A046927, A194831 (size of records).

ways[n_] := ways[n] = (w = 0; Do[ p = 2k + 1; q = n - k; If[PrimeQ[p] && PrimeQ[q], w++], {k, 1, n}]; w); record = 0; A194830 = Reap[Do[If[ways[n] > record, record = ways[n]; Print["2n+1 = ", 2n + 1, " record = ", record];  Sow[{ways[n], 2n + 1}]], {n, 0, 12000}]][[2, 1]][[All, 2]] (* Jean-François Alcover, Dec 05 2012 *)

cp's OEIS Frontend

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

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