OFFSET
1,1
COMMENTS
The matching values of k are given by A103183.
Note that 1/20 + 1/50 = 0.070 --> 70 is a "sliding number" though 0.070 is usually written as 0.07. So any "sliding number" G produces an infinite chain of others: G*10, G*100, G*1000, etc. The first number in such a chain is called "primitive". [Note the sequence is not defined by way of decimal digits. The power-of-10 relationship merely makes it practical to illustrate examples using decimal fractions. - Peter Munn, Nov 29 2022]
Another characterization of the sliding numbers is numbers of the form j + 10^k/j where j | 10^k (k >= 0). - David W. Wilson, Mar 09 2005
In the definition 1/r + 1/s is (r+s)/(r*s) and we can cancel out (r+s), so we are looking for all (r+s) with r and s positive integers, such that r*s is a (nonnegative integral) power of 10, sorted by (r+s). It follows, for example, that for a given k, the smallest (r+s) that is a solution is at least 2*sqrt(10^k) (equality when k is even, unless we also require r != s). - Christopher Landauer, Mar 10 2005
Equivalently, the terms are numbers with a partition into 2 integers whose product is a power of 10. - Peter Munn, Nov 28 2022
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Giovanni Resta, Sliding numbers
EXAMPLE
1/4 + 1/25 = 0.29 --> 29 is a "sliding number"
1/8 + 1/125 = 0.133 --> 133 is a "sliding number"
1/2 + 1/5 = 0.7 --> 7 is a "sliding number"
From Hugo Pfoertner and Michael De Vlieger, Nov 28 2022: (Start)
Terms are repeated if they can be represented in different ways, for example:
205 = 5 + 200; 5 * 200 = 1000 (so 1/5 + 1/200 = 205/1000)
= 80 + 125; 80 * 125 = 10000 (so 1/80 + 1/125 = 205/10000).
(End)
MATHEMATICA
nn = 6; f[n_] := Transpose@ {Reverse@ #[[1 ;; Ceiling[Length[#]/2]]], #[[-Ceiling[Length[#]/2] ;; -1]]} &@ Divisors[10^n]; TakeWhile[Sort@ Flatten@ Table[Total /@ f[k], {k, 0, nn}], # < Total@ First@ f[nn + 1] &]] (* Michael De Vlieger, Oct 08 2022 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Eric Angelini, Mar 09 2005
EXTENSIONS
More terms from Kerry Mitchell, Mar 09 2005
Edited by Peter Munn, Nov 27 2022
STATUS
approved