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 A304034 Number of ways to write n as p + 2^k + (1+(n mod 2))*3^m with p prime, where k and m are positive integers with 2^k + (1+(n mod 2))*3^m squarefree. 16
 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 5, 1, 3, 2, 5, 1, 7, 3, 3, 4, 4, 4, 6, 2, 3, 5, 6, 2, 7, 3, 5, 5, 6, 5, 9, 3, 4, 6, 7, 2, 12, 2, 5, 6, 7, 4, 10, 3, 3, 5, 8, 2, 8, 3, 4, 6, 8, 5, 9, 4, 2, 7, 7, 3, 13, 5, 5, 9, 7, 5, 13, 3, 6, 10, 7, 5, 10, 5, 7, 7, 9, 8, 13 (list; graph; refs; listen; history; text; internal format)
 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. Sequence in context: A029184 A179049 A029221 * A029183 A213423 A265753 Adjacent sequences:  A304031 A304032 A304033 * A304035 A304036 A304037 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 06 2018 STATUS approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)