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

%I A007629 M4922 #104 May 21 2025 18:56:09
%S A007629 14,19,28,47,61,75,197,742,1104,1537,2208,2580,3684,4788,7385,7647,
%T A007629 7909,31331,34285,34348,55604,62662,86935,93993,120284,129106,147640,
%U A007629 156146,174680,183186,298320,355419,694280,925993,1084051,7913837,11436171,33445755,44121607
%N A007629 Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).
%C A007629 Numbers k > 9 with following property: form a sequence {b(i)} whose initial terms are the t digits of k, later terms given by rule that b(i) = sum of t previous terms; then k itself appears in the sequence.
%C A007629 Called Sep-Numbers by Baumann (2004). - _N. J. A. Sloane_, Mar 02 2014
%C A007629 Sometimes named after the American mathematician, software engineer and author Mike Keith (b. 1955), who introduced them in 1987 as "repfigit numbers". - _Amiram Eldar_, Jun 27 2021
%D A007629 Charles Ashbacher, J. Rec. Math., Vol. 21, No. 4 (1989), p. 310.
%D A007629 Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 197, p. 59, Ellipses, Paris 2008.
%D A007629 Mike Keith, Repfigit Numbers, J. Recreational Math., Vol. 19, No. 2 (1987), pp. 41-42.
%D A007629 Clifford A. Pickover, All Known Replicating Fibonacci Digits Less Than One Billion, J. Recreational Math., Vol. 22, No. 3, p. 176, 1990.
%D A007629 Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 229.
%D A007629 Clifford A. Pickover, Wonders of Numbers, "Looping Replicating Fibonacci digits", pp. 174-5, OUP 2000.
%D A007629 K. Sherriff, Computing Replicating Fibonacci Digits, J. Recreational Math., Vol. 26, No. 3, p. 191, 1994.
%D A007629 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007629 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, see p. 71.
%H A007629 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 A007629 Rüdeger Baumann, <a href="http://www.austromath.at/dug/dnl53.pdf">Sep-Zahlen or Sep-Numbers</a>, DERIVE Newsletter, #53 (2004), p. 33.
%H A007629 Jhon J. Bravo, Sergio Guzmán, and Florian Luca, <a href="http://dx.doi.org/10.1007/s10986-013-9199-3">Repdigit Keith numbers</a>, Lithuanian Mathematical Journal, Vol. 53, No. 2 (April 2013), pp. 143-148.
%H A007629 Edmund Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=uuMwz47LV_w">Keith Numbers</a>, Numberphile video (2012).
%H A007629 Mike Keith, <a href="http://www.cadaeic.net/keithnum.htm">Keith numbers</a>.
%H A007629 Mike 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 A007629 Mike Keith, <a href="/A007629/a007629.pdf">Power-sum numbers</a>, J. Recreational Mathematics, Vol. 18, No. 4 (1986), pp. 275-278. (Annotated scanned copy)
%H A007629 Martin Klazar and Florian Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Klazar/klazar15.html">Counting Keith numbers</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.2.
%H A007629 Madras Math's Amazing Number Facts, <a href="http://www.madras.fife.sch.uk/maths/amazingnofacts/fact049.html">Repfigits</a>.
%H A007629 Clifford 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 A007629 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KeithNumber.html">Keith Number</a>.
%H A007629 Wikipedia, <a href="http://en.wikipedia.org/wiki/Keith_number">Keith number</a>.
%e A007629 197 is a term since sequence {b(i)} (see Comments) is A186830 = { 1, 9, 7, 17, 33, 57, 107, 197, ... }, which contains 197.
%p A007629 isA007629 := proc(n)
%p A007629     local L,t,a ;
%p A007629     if n < 10 then
%p A007629         return false;
%p A007629     end if;
%p A007629     L := ListTools[Reverse](convert(n,base,10)) ;
%p A007629     t :=  nops(L) ;
%p A007629     while true do
%p A007629         a := add(op(-i,L),i=1..t) ;
%p A007629         L := [op(L),a] ;
%p A007629         if a > n then
%p A007629             return false;
%p A007629         elif a = n then
%p A007629             return true;
%p A007629         end if;
%p A007629     end do:
%p A007629 end proc:
%p A007629 for n from 1 do
%p A007629     if isA007629(n) then
%p A007629         printf("%d,\n",n);
%p A007629     end if;
%p A007629 end do: # _R. J. Mathar_, Jan 12 2016
%t A007629 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 A007629 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 *)
%t A007629 keithQ[n_, e_] := Last[NestWhile[Rest[Append[#, Apply[Plus, #]]]&, IntegerDigits[n^e], Last[#]<n&]]==n/;n>9
%t A007629 a007629[n_] := Select[Range[10, n], keithQ[#, 1]&]
%t A007629 a007629[45*10^6] (* _Hartmut F. W. Hoft_, Jun 02 2021 *)
%o A007629 (Haskell)
%o A007629 import Data.Char (digitToInt)
%o A007629 a007629 n = a007629_list !! (n-1)
%o A007629 a007629_list = filter isKeith [10..] where
%o A007629    isKeith n = repfigit $ reverse $ map digitToInt $ show n where
%o A007629       repfigit ns = s == n || s < n && (repfigit $ s : init ns) where
%o A007629          s = sum ns
%o A007629 -- _Reinhard Zumkeller_, Nov 04 2010, Mar 31 2011
%o A007629 (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 A007629 (Python)
%o A007629 A007629_list = []
%o A007629 for n in range(10,10**9):
%o A007629     x = [int(d) for d in str(n)]
%o A007629     y = sum(x)
%o A007629     while y < n:
%o A007629         x, y = x[1:]+[y], 2*y-x[0]
%o A007629     if y == n:
%o A007629         A007629_list.append(n) # _Chai Wah Wu_, Sep 12 2014
%Y A007629 Cf. A006576, A048970, A050235, A186830. See A130010 for another version.
%Y A007629 Cf. A162724, A187713, A188195-A188200 (base 2, 5, 3-4, 6-9).
%Y A007629 Cf. A188380 (balanced ternary), A188381 (base -2).
%Y A007629 Cf. A188201 (least base-n Keith number >= n).
%Y A007629 Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920, A281921 (starting with n^k, 2<=k<=10).
%K A007629 nonn,base,nice
%O A007629 1,1
%A A007629 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E A007629 12th term corrected from 2508 to 2580, Aug 15 1997
%E A007629 More terms from _Mike Keith_, Feb 15 1999
%E A007629 Keith's old links fixed and C. Ashbacher's name added by _Christopher Carl Heckman_, Nov 18 2010