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%I #49 Aug 09 2024 10:08:49
%S 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,
%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,
%U 52,53,54,55,56,57,58,59,60,61,62,63,65,66,67
%N 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.
%C 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
%C Or exponentially Fibonacci numbers. - _Vladimir Shevelev_, Nov 15 2015
%C Sequences A004709, A005117, A046100 are subsequences. - _Vladimir Shevelev_, Nov 16 2015
%C 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
%C Numbers whose sets of unitary divisors (A077610) and Zeckendorf-infinitary divisors (see A318465) coincide. Also, numbers whose sets of unitary divisors and dual-Zeckendorf-infinitary divisors (see A331109) coincide. - _Amiram Eldar_, Aug 09 2024
%H Amiram Eldar, <a href="/A115063/b115063.txt">Table of n, a(n) for n = 1..10000</a>
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1510.05914">Exponentially S-numbers</a>, arXiv:1510.05914 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1511.03860">Set of all densities of exponentially S-numbers</a>, arXiv:1511.03860 [math.NT], 2015.
%H Vladimir Shevelev, <a href="http://dx.doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175 (2016), 385-395.
%F 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_.
%F For a proof of the formula, see [Shevelev], the first link. - _Vladimir Shevelev_, Nov 17 2015
%e 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
%t 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 *)
%Y Cf. A004709, A005117, A046100, A077610, A115975, A197680, A209061, A318465, A331109.
%K easy,nonn
%O 1,2
%A _Giovanni Teofilatto_, Mar 01 2006
%E a(35) inserted by _Amiram Eldar_, Oct 06 2019