

A303949


Number of ways to write 2*n+1 as p + 2*(2^k+5^m) with p prime and 2^k+5^m a product of at most three distinct primes, where k and m are nonnegative integers.


5



0, 0, 1, 2, 2, 2, 3, 5, 3, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6, 4, 3, 6, 7, 3, 6, 9, 7, 5, 8, 7, 6, 7, 9, 7, 8, 2, 8, 9, 5, 5, 6, 6, 7, 6, 6, 7, 10, 6, 7, 9, 5, 6, 8, 6, 3, 6, 7, 7, 8, 5, 10, 9, 8, 5, 9, 5, 7, 10, 5, 4, 10, 7, 6, 8, 6, 7, 8, 7, 6, 8, 6
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OFFSET

1,4


COMMENTS

4787449 is the first value of n > 2 with a(n) = 0, and 2*4787449+1 = 9574899 has the unique representation as p + 2*(2^k+5^m): 9050609 + 2*(2^18+5^0) with 9050609 prime and 2^18+5^0 = 5*13*37*109.
See also A303934 and A304081 for related conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341353, Springer, New York, 2010.
ZhiWei 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. 279310. (See also arXiv:1211.1588 [math.NT], 20122017.)


EXAMPLE

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


MATHEMATICA

qq[n_]:=qq[n]=SquareFreeQ[n]&&Length[FactorInteger[n]]<=3;
tab={}; Do[r=0; Do[If[SquareFreeQ[2^k+5^m]&&PrimeQ[2n+12(2^k+5^m)], r=r+1], {k, 0, Log[2, n]}, {m, 0, Log[5, n+1/22^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, A303934, A304031, A304032, A304081.
Sequence in context: A228405 A276824 A235998 * A333462 A133850 A160902
Adjacent sequences: A303946 A303947 A303948 * A303950 A303951 A303952


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 05 2018


STATUS

approved



