A303934 Number of ways to write 2*n as p + 2^k + 5^m with p prime and 2^k + 5^m squarefree, where k and m are nonnegative integers.

0, 1, 1, 3, 3, 2, 2, 3, 3, 4, 3, 5, 4, 4, 3, 4, 5, 7, 4, 7, 4, 8, 7, 6, 7, 6, 5, 5, 5, 7, 5, 8, 5, 5, 8, 6, 9, 9, 6, 8, 6, 6, 7, 8, 4, 7, 8, 7, 3, 10, 6, 7, 8, 7, 7, 9, 5, 8, 7, 6, 5, 5, 6, 3, 11, 7, 9, 12, 8, 12, 10, 11, 11, 9, 7, 9, 7, 8, 8, 11, 7, 11, 8, 9, 15, 11, 8, 9, 8, 9

Offset

1,4

Comments

Conjecture: a(n) > 0 for all n > 1.

This has been verified for all n = 2..10^10.

Note that a(n) <= A303821(n).

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 + 5^0 with 2 prime and 2^0 + 5^0 squarefree.
a(3) = 1 since 2*3 = 3 + 2^1 + 5^0 with 3 prime and 2^1 + 5^0 squarefree.

Mathematica

tab={}; Do[r=0; Do[If[SquareFreeQ[2^k+5^m]&&PrimeQ[2n-2^k-5^m], r=r+1], {k, 0, Log[2, 2n-1]}, {m, 0, Log[5, 2n-2^k]}]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]

Crossrefs

Cf. A000040, A000079, A000351, A005117, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A304034, A304081, A304122.

Sequence in context: A075788 A362503 A324080 * A304031 A162235 A113780

Adjacent sequences: A303931 A303932 A303933 * A303935 A303936 A303937

Keyword

nonn

Author

Zhi-Wei Sun, May 03 2018

Status

approved