

A130792


Numbers n whose representation can be split in two parts which can be used as seeds for a Fibonaccilike sequence containing n itself.


2



14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995
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OFFSET

1,1


COMMENTS

The 6 members with two digits are also Keith numbers. There are 233 numbers below 10^6 in this sequence.
If the number x is rewritten as a U b, the problem is to find a value of y such that x = a*F(y) + b*F(y+1), where F(y) is a Fibonacci number (see file with values of x, a, b, y, for 1<x<10^6, in Links). All the listed numbers admit only one concatenation, a U b, that, through the addition process, leads to themselves. Is there any number that admits more than one single concatenation? [Paolo P. Lava, Oct 02 2014]
Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as 1 U 9, 1 U 99, 1 U 999, 1 U 9...9. In two steps we have the initial numbers back: 1 + 9 = 10 and 9 + 10 = 19; 1 + 99 = 100 and 99 + 100 = 199, etc. [Paolo P. Lava, Oct 08 2014]


LINKS

Paolo Lava and Michel Marcus, Table of n, a(n) for n = 1..406 (first 200 terms from Paolo Lava)
Paolo P. Lava, List of [x,a,b,y] in the equation x = a*F(y) + b*F(y+1), for 1< x <10^6


EXAMPLE

122 can be split into 12 and 2 and the Fibonaccilike sequence: 12, 2, 14, 16, 30, 46, 76, 122, ... contains 122 itself.


MATHEMATICA

testQ[n_]:= Block[{x, y, z, p = 10, r = False}, While[p < n, x = Floor[n/p]; y = Mod[n, p]; While[y < n, z = x + y; x = y; y = z]; If[y == n, r = True; Break[]]; p *= 10]; r]; Select[Range[10^4], testQ]


PROG

(PARI) isok(n) = {nb = #Str(n); for (i=1, nb1, x = n\10^i; y = n  10^i*x; ok = 0; while(!ok, z = x + y; if (z > n, ok = 1); if (z == n, return (1)); x = y; y = z; )); } \\ Michel Marcus, Oct 08 2014


CROSSREFS

Cf. A007629.
Sequence in context: A028396 A120158 A292614 * A121235 A007629 A241199
Adjacent sequences: A130789 A130790 A130791 * A130793 A130794 A130795


KEYWORD

base,nonn


AUTHOR

Giovanni Resta, Aug 20 2007


STATUS

approved



