OFFSET
1,10
COMMENTS
Conjecture: a(n) > 0 for all n > 11.
This has been verified for n up to 10^10.
See also A304081 for a similar conjecture.
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(8) = 1 since 8 = 3 + 2^1 + 3^1 with 3 prime and 2^1 + 3^1 = 5 squarefree.
a(13) = 1 since 13 = 3 + 2^2 + 2*3^1 with 3 prime and 2^2 + 2*3^1 = 2*5 squarefree.
a(19) = 1 since 19 = 5 + 2^3 + 2*3^1 with 5 prime and 2^3 + 2*3^1 = 2*7 squarefree.
a(23) = 1 since 23 = 13 + 2^2 + 2*3^1 with 13 prime and 2^2 + 2*3 = 2*5 squarefree.
MATHEMATICA
tab={}; Do[r=0; Do[If[SquareFreeQ[2^k+(1+Mod[n, 2])*3^m]&&PrimeQ[n-2^k-(1+Mod[n, 2])*3^m], r=r+1], {k, 1, Log[2, n]}, {m, 1, If[2^k==n, -1, Log[3, (n-2^k)/(1+Mod[n, 2])]]}]; tab=Append[tab, r], {n, 1, 90}]; 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, A304032, A304081.
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 06 2018
STATUS
approved