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.

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

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

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

Cf. A004709, A005117, A046100, A077610, A115975, A197680, A209061, A318465, A331109.

Sequence in context: A194897 A140823 A209061 * A369939 A178210 A013938

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