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

1,3

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

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.)

Example

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.

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]

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.

Sequence in context: A001414 A134875 A134889 * A319057 A181894 A265535

Adjacent sequences: A303699 A303700 A303701 * A303703 A303704 A303705

Keyword

nonn

Author

Zhi-Wei Sun, Apr 29 2018

Status

approved