

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
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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_keps_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 Snumbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially Snumbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, Sexponential numbers, Acta Arithmetica, Vol. 175(2016), 385395.


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(i1))/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 AriasdeReyna 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



