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

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

Author

Hugo Pfoertner, Sep 12 2011

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• See A046927.

Links

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

Crossrefs

Cf. A046927, A194831 (size of records).

Programs

Mathematica

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 *)