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A007629 Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).
(Formerly M4922)
27
14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171, 33445755, 44121607 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

Called Sep-Numbers by Baumann (2004). - N. J. A. Sloane, Mar 02 2014

REFERENCES

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

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

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

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

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

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

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

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

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..94 [Taken from first Keith link below.]

Rüdeger Baumann, Sep-Zahlen or Sep-Numbers, DERIVE Newsletter, #53 (2004), p. 33.

Jhon J. Bravo, Sergio Guzmán, Florian Luca, Repdigit Keith numbers, Lithuanian Mathematical Journal, April 2013, Volume 53, Issue 2, pp 143-148.

M. Keith, Keith numbers

M. Keith, Determination of All Keith Numbers Up to 10^19.

M. Klazar and F. Luca, Counting Keith numbers, Journal of Integer Sequences, Vol. 10 (2007), #07.2.2.

Madras Math's Amazing Number Facts, Repfigits

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Eric Weisstein's World of Mathematics, Keith Number.

Wikipedia, Keith number

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

import Data.Char (digitToInt

a007629 n = a007629_list !! (n-1)

a007629_list = filter isKeith [10..] where

   isKeith n = repfigit $ reverse $ map digitToInt $ show n where

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

         s = sum ns

-- Reinhard Zumkeller, Nov 04 2010, Mar 31 2011

(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

(Python)

A007629_list = []

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

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

....y = sum(x)

....while y < n:

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

....if y == n:

........A007629_list.append(n) # Chai Wah Wu, Sep 12 2014

CROSSREFS

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

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

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

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

Sequence in context: A120158 A130792 A121235 * A241199 A092768 A144080

Adjacent sequences:  A007626 A007627 A007628 * A007630 A007631 A007632

KEYWORD

nonn,base,nice,changed

AUTHOR

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

EXTENSIONS

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

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

Keith's old links fixed and C. Ashbacher's name added by Christopher Carl Heckman, Nov 18 2010

STATUS

approved

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Last modified September 17 12:52 EDT 2014. Contains 246844 sequences.