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
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
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
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
KEYWORD
easy,nonn
AUTHOR
Giovanni Teofilatto, Mar 01 2006
EXTENSIONS
a(35) inserted by Amiram Eldar, Oct 06 2019
STATUS
approved