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 A115063 Natural numbers of the form p^F(n_p)*q^F(n_q)*r^F(n_r)*...*z^F(n_z), where p,q,r,... are distinct primes and F(n) is a Fibonacci number. 4
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The complementary sequence is 16, 48, 64, 80, 81, 112, 128, 144, 162, 176, 192, 208, 240, 272, 304, 320, 324, 336, 368, 384, 400, ... - R. J. Mathar, Apr 22 2010 Or exponentially Fibonacci numbers. - Vladimir Shevelev, Nov 15 2015 Sequences A004709, A005117, A046100 are subsequences. - Vladimir Shevelev, Nov 16 2015 Let h_k be the density of the subsequence of A115063 of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_k-eps_k, h_k) there is no a sequence S of positive integers such that x is the density of numbers whose prime power factorization has the form Product_i p_i^e_i where the e_i are all in S. For a proof, see [Shevelev], the second link. - Vladimir Shevelev, Nov 17 2015 LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015. Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015. Vladimir  Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395. FORMULA Sum_{i<=x, i is in A115063} 1 = h*x+O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c=4*sqrt(2.4/log 2)=7.44308... and h=Product_(prime p)(1+Sum_(i>=2)(u(i)-u(i-1))/p^i)=0.944335905... where u(n)is the characteristic function of sequence A000045. The calculations of h over the formula were done independently by Juan Arias-de-Reyna and Peter J. C. Moses. For a proof of the formula, see [Shevelev], the first link. - Vladimir Shevelev, Nov 17 2015 EXAMPLE 12 is a term, since 12=2^2*3^1 and the exponents 2 and 1 are terms of Fibonacci sequence (A000045). - Vladimir Shevelev, Nov 15 2015 MATHEMATICA fibQ[n_] := IntegerQ @ Sqrt[5 n^2 - 4] || IntegerQ @ Sqrt[5 n^2 + 4]; aQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], fibQ]; Select[Range[100], aQ] (* Amiram Eldar, Oct 06 2019 *) CROSSREFS Cf. A004709, A005117, A046100, A197680, A209061. Sequence in context: A194897 A140823 A209061 * A178210 A013938 A023809 Adjacent sequences:  A115060 A115061 A115062 * A115064 A115065 A115066 KEYWORD easy,nonn AUTHOR Giovanni Teofilatto, Mar 01 2006 EXTENSIONS a(35) inserted by Amiram Eldar, Oct 06 2019 STATUS approved

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Last modified January 29 14:07 EST 2020. Contains 331338 sequences. (Running on oeis4.)