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 A007629 Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers). (Formerly M4922) 55

%I M4922

%S 14,19,28,47,61,75,197,742,1104,1537,2208,2580,3684,4788,7385,7647,

%T 7909,31331,34285,34348,55604,62662,86935,93993,120284,129106,147640,

%U 156146,174680,183186,298320,355419,694280,925993,1084051,7913837,11436171,33445755,44121607

%N Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).

%C Numbers n>9 with following property: form a sequence b(i) whose initial terms are the t digits of n, later terms given by rule that b(i) = sum of t previous terms; then n itself appears in the sequence.

%C Called Sep-Numbers by Baumann (2004). - _N. J. A. Sloane_, Mar 02 2014

%D C. Ashbacher, J. Rec. Math., vol. 21, no. 4, p. 310, 1989.

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 197, p. 59, Ellipses, Paris 2008.

%D M. Keith, Repfigit Numbers, J. Recreational Math., Vol. 19, No. 2, pp. 41-42, 1987.

%D C. A. Pickover, All Known Replicating Fibonacci Digits Less Than One Billion, J. Recreational Math., Vol. 22, No. 3, p. 176, 1990.

%D C. A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 229.

%D C. A. Pickover, Wonders of Numbers, "Looping Replicating Fibonacci digits", pp. 174-5, OUP 2000.

%D K. Sherriff, Computing Replicating Fibonacci Digits, J. Recreational Math., Vol. 26, No. 3, p. 191, 1994.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, see p. 71.

%H N. J. A. Sloane, <a href="/A007629/b007629.txt">Table of n, a(n) for n = 1..94</a> [Taken from first Keith link below.]

%H Rüdeger Baumann, <a href="http://www.austromath.at/dug/dnl53.pdf">Sep-Zahlen or Sep-Numbers</a>, DERIVE Newsletter, #53 (2004), p. 33.

%H Jhon J. Bravo, Sergio Guzmán, Florian Luca, <a href="http://dx.doi.org/10.1007/s10986-013-9199-3">Repdigit Keith numbers</a>, Lithuanian Mathematical Journal, April 2013, Volume 53, Issue 2, pp 143-148.

%H M. Keith, <a href="http://www.cadaeic.net/keithnum.htm">Keith numbers</a>

%H M. Keith, <a href="http://web.archive.org/web/20070208082537/http://users.aol.com/s6sj7gt/keithnum.htm">Determination of All Keith Numbers Up to 10^19.</a>

%H M. Klazar and F. Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Klazar/klazar15.html">Counting Keith numbers</a>, Journal of Integer Sequences, Vol. 10 (2007), #07.2.2.

%H Madras Math's Amazing Number Facts, <a href="http://www.madras.fife.sch.uk/maths/amazingnofacts/fact049.html">Repfigits</a>

%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KeithNumber.html">Keith Number.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Keith_number">Keith number</a>

%e 197 is a term since sequence is 1, 9, 7, 17, 33, 57, 107, 197, ..., which contains 197.

%p isA007629 := proc(n)

%p local L,t,a ;

%p if n < 10 then

%p return false;

%p end if;

%p L := ListTools[Reverse](convert(n,base,10)) ;

%p t := nops(L) ;

%p while true do

%p a := add(op(-i,L),i=1..t) ;

%p L := [op(L),a] ;

%p if a > n then

%p return false;

%p elif a = n then

%p return true;

%p end if;

%p end do:

%p end proc:

%p for n from 1 do

%p if isA007629(n) then

%p printf("%d,\n",n);

%p end if;

%p end do: # _R. J. Mathar_, Jan 12 2016

%t keithQ[n_Integer] := Module[{b = IntegerDigits[n], s, k = 0}, s = Total[b]; While[s < n, AppendTo[b, s]; k++; s = 2*s - b[[k]]]; s == n]; Select[Range[10, 100000], keithQ] (* _T. D. Noe_, Mar 15 2011 *)

%t nxt[n_]:=Rest[Flatten[Join[{n,Total[n]}]]]; repfigitQ[m_]:=MemberQ[ NestWhileList[ nxt,IntegerDigits[m],Max[#]<=m&][[All,-1]],m]; Select[ Range[10,45*10^6],repfigitQ] (* _Harvey P. Dale_, Sep 02 2016 *)

%o import Data.Char (digitToInt

%o a007629 n = a007629_list !! (n-1)

%o a007629_list = filter isKeith [10..] where

%o isKeith n = repfigit \$ reverse \$ map digitToInt \$ show n where

%o repfigit ns = s == n || s < n && (repfigit \$ s : init ns) where

%o s = sum ns

%o -- _Reinhard Zumkeller_, Nov 04 2010, Mar 31 2011

%o (PARI) is(n)=if(n<14,return(0));my(v=digits(n),t=#v);while(v[#v]<n,v=concat(v,sum(i=0,t-1,v[#v-i]))); v[#v]==n \\ _Charles R Greathouse IV_, Feb 01 2013

%o (Python)

%o A007629_list = []

%o for n in range(10,10**9):

%o ....x = [int(d) for d in str(n)]

%o ....y = sum(x)

%o ....while y < n:

%o ........x, y = x[1:]+[y], 2*y-x[0]

%o ....if y == n:

%o ........A007629_list.append(n) # _Chai Wah Wu_, Sep 12 2014

%Y Cf. A006576, A048970, A050235, A186830. See A130010 for another version.

%Y Cf. A162724, A187713, A188195-A188200 (base 2, 5, 3-4, 6-9).

%Y Cf. A188380 (balanced ternary), A188381 (base -2).

%Y Cf. A188201 (least base-n Keith number >= n).

%K nonn,base,nice

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_

%E 12th term corrected from 2508 to 2580, Aug 15 1997

%E More terms from Mike Keith (Domnei(AT)aol.com), Feb 15 1999

%E Keith's old links fixed and C. Ashbacher's name added by _Christopher Carl Heckman_, Nov 18 2010

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Last modified May 27 09:52 EDT 2017. Contains 287204 sequences.