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%I #35 Nov 29 2022 12:15:34
%S 2,7,11,20,25,29,52,65,70,101,110,133,200,205,205,250,254,290,425,502,
%T 520,641,650,700,785,925,1001,1010,1100,1258,1330,2000,2005,2050,2050,
%U 2225,2500,2504,2540,2900,3157,3445,4025,4250,5002,5020
%N Sliding numbers: totals, including repetitions, of sums r + s, r >= s, such that 1/r + 1/s = (r + s)/10^k for some k >= 0.
%C The matching values of k are given by A103183.
%C 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]
%C 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
%C 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
%C Equivalently, the terms are numbers with a partition into 2 integers whose product is a power of 10. - _Peter Munn_, Nov 28 2022
%H Michael De Vlieger, <a href="/A103182/b103182.txt">Table of n, a(n) for n = 1..10000</a>
%H Giovanni Resta, <a href="https://www.numbersaplenty.com/set/sliding_number/">Sliding numbers</a>
%e 1/4 + 1/25 = 0.29 --> 29 is a "sliding number"
%e 1/8 + 1/125 = 0.133 --> 133 is a "sliding number"
%e 1/2 + 1/5 = 0.7 --> 7 is a "sliding number"
%e From _Hugo Pfoertner_ and _Michael De Vlieger_, Nov 28 2022: (Start)
%e Terms are repeated if they can be represented in different ways, for example:
%e 205 = 5 + 200; 5 * 200 = 1000 (so 1/5 + 1/200 = 205/1000)
%e = 80 + 125; 80 * 125 = 10000 (so 1/80 + 1/125 = 205/10000).
%e (End)
%t 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 *)
%Y Cf. A103183, A103184.
%K nonn,base
%O 1,1
%A _Eric Angelini_, Mar 09 2005
%E More terms from _Kerry Mitchell_, Mar 09 2005
%E Edited by _Peter Munn_, Nov 27 2022
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