A303702 Number of ways to write 2*n as p + 2^k + 3^m, where p is a prime, and k and m are nonnegative integers.

0, 1, 2, 3, 4, 5, 5, 7, 6, 6, 9, 9, 5, 8, 9, 6, 9, 11, 8, 10, 11, 7, 12, 15, 8, 10, 12, 7, 10, 9, 8, 12, 11, 5, 12, 16, 7, 13, 17, 8, 10, 15, 10, 13, 14, 10, 12, 17, 7, 12, 18, 11, 13, 17, 10, 13, 20, 11, 14, 17, 8, 10, 16, 7, 10

Offset: 1

Author

Zhi-Wei Sun, Apr 29 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1. In other words, any even number greater than 2 can be written as the sum of a prime, a power of 2 and a power of 3.

It has been verified that a(n) > 0 for all n = 2..3*10^9.

a(n) > 0 for n <= 10^11. - Jud McCranie, Jun 25 2023

a(n) > 0 for n < 10^12. - Jud McCranie, Jul 11 2023

a(n) > 0 for n <= 4*10^12. - Jud McCranie, Aug 17 2023

Examples

			a(2) = 1 since 2*2 = 2 + 2^0 + 3^0 with 2 prime.
a(3) = 2 since 2*3 = 2 + 2^0 + 3^1 = 3 + 2^1 + 3^0 with 2 and 3 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000040, A000079, A000244, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660.

Programs

Mathematica

tab={};Do[r=0;Do[If[PrimeQ[2n-2^x-3^y],r=r+1],{x,0,Log[2,2n-1]},{y,0,Log[3,2n-2^x]}];tab=Append[tab,r],{n,1,65}];Print[tab]