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 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 (list; graph; refs; listen; history; text; internal format)
 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 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+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+1-2(2^k+5^m)], r=r+1], {k, 0, Log[2, n]}, {m, 0, Log[5, n+1/2-2^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, A303934, A304031, A304032, A304081. Sequence in context: A228405 A276824 A235998 * A333462 A133850 A160902 Adjacent sequences:  A303946 A303947 A303948 * A303950 A303951 A303952 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 05 2018 STATUS approved

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Last modified September 22 09:32 EDT 2020. Contains 337289 sequences. (Running on oeis4.)