A304032 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

A304032

Number of ways to write 2*n as p + 2^k + 3^m with p prime and 2^k + 3^m a product of at most two distinct primes, where k and m are nonnegative integers.

0, 1, 1, 3, 4, 4, 4, 6, 6, 5, 8, 9, 4, 6, 7, 4, 9, 10, 6, 9, 10, 6, 11, 14, 7, 9, 11, 5, 10, 9, 6, 12, 10, 3, 11, 15, 7, 12, 16, 7, 9, 14, 9, 12, 14, 8, 12, 16, 5, 12, 18, 10, 12, 16, 9, 12, 19, 10, 13, 17, 6, 10, 15, 6, 10, 16, 10, 12, 15, 10, 17, 20, 8, 14, 15, 8, 11, 18, 9, 12

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The even number 58958 cannot be written as p + 2^k + 3^m with p and 2^k + 3^m both prime.

Clearly, a(n) <= A303702(n). We note that a(n) > 0 for all n = 2..5*10^8.

See also A304034 for a related conjecture.

REFERENCES

J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16(1973), 157-176.

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

MATHEMATICA

qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=2;

tab={}; Do[r=0; Do[If[qq[2^k+3^m]&&PrimeQ[2n-2^k-3^m], r=r+1], {k, 0, Log[2, 2n-1]}, {m, 0, Log[3, 2n-2^k]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

CROSSREFS

Cf. A000040, A000079, A000224, A005117, A118955, A155216, 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, A303934, A303949, A304031, A304034, A304081.

Sequence in context: A179842 A259052 A268395 * A022484 A361497 A352717

Adjacent sequences: A304029 A304030 A304031 * A304033 A304034 A304035

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 04 2018

STATUS

approved