A162724 -id:A162724 - cp's OEIS frontend

A265426 Primes p such that p - 1 is a binary Keith number (A162724).

2, 3, 5, 17, 257, 1367, 65537, 2960687

Offset: 1

Jaroslav Krizek, Dec 08 2015

See A162724 (binary Keith numbers) and A007629 (Keith numbers) for definitions.
Primes of the form A162724(n)+1.
Fermat primes (A019434) are terms.
The next term, if it exists, must be greater than 17*10^9.
Union of primes p of the form A162363(n)+1 and A000079(m)+1 for a some n or m.

Cf. A000079, A007629, A019434, A162363, A162724.

fQ[n_] := Module[{b = IntegerDigits[n, 2], s}, s = Total@ b; If[s <= 1, True, k = 1; While[s = 2 s - b[[k]]; s < n, k++]; s == n]]; Select[Prime@ Range[10^6], fQ[# - 1] &] (* Michael De Vlieger, Dec 09 2015, after T. D. Noe at A162724 *)

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

Original entry on oeis.org

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

Offset: 1

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

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

			197 is a term since sequence {b(i)} (see Comments) is A186830 = { 1, 9, 7, 17, 33, 57, 107, 197, ... }, which contains 197.

Charles Ashbacher, J. Rec. Math., Vol. 21, No. 4 (1989), p. 310.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 197, p. 59, Ellipses, Paris 2008.
Mike Keith, Repfigit Numbers, J. Recreational Math., Vol. 19, No. 2 (1987), pp. 41-42.
Clifford A. Pickover, All Known Replicating Fibonacci Digits Less Than One Billion, J. Recreational Math., Vol. 22, No. 3, p. 176, 1990.
Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 229.
Clifford 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.

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, and Florian Luca, Repdigit Keith numbers, Lithuanian Mathematical Journal, Vol. 53, No. 2 (April 2013), pp. 143-148.
Edmund Copeland and Brady Haran, Keith Numbers, Numberphile video (2012).
Mike Keith, Keith numbers.
Mike Keith, Determination of All Keith Numbers Up to 10^19.
Mike Keith, Power-sum numbers, J. Recreational Mathematics, Vol. 18, No. 4 (1986), pp. 275-278. (Annotated scanned copy)
Martin Klazar and Florian Luca, Counting Keith numbers, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.2.
Madras Math's Amazing Number Facts, Repfigits.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Eric Weisstein's World of Mathematics, Keith Number.
Wikipedia, Keith number.

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).
Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920, A281921 (starting with n^k, 2<=k<=10).

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

isA007629 := proc(n)
    local L,t,a ;
    if n < 10 then
        return false;
    end if;
    L := ListTools[Reverse](convert(n,base,10)) ;
    t :=  nops(L) ;
    while true do
        a := add(op(-i,L),i=1..t) ;
        L := [op(L),a] ;
        if a > n then
            return false;
        elif a = n then
            return true;
        end if;
    end do:
end proc:
for n from 1 do
    if isA007629(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

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 *)
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 *)
keithQ[n_, e_] := Last[NestWhile[Rest[Append[#, Apply[Plus, #]]]&, IntegerDigits[n^e], Last[#]9
a007629[n_] := Select[Range[10, n], keithQ[#, 1]&]
a007629[45*10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

is(n)=if(n<14,return(0));my(v=digits(n),t=#v);while(v[#v]Charles R Greathouse IV, Feb 01 2013

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

12th term corrected from 2508 to 2580, Aug 15 1997
More terms from Mike Keith, Feb 15 1999
Keith's old links fixed and C. Ashbacher's name added by Christopher Carl Heckman, Nov 18 2010

A187713 Base-5 Keith numbers.

Original entry on oeis.org

5, 9, 10, 11, 13, 15, 20, 22, 31, 40, 43, 53, 62, 71, 84, 93, 124, 154, 221, 483, 3044, 18748, 125973, 232085, 1705260, 2091605, 5616236, 8067806, 8849508, 58944155, 84572166, 164487062, 421825427, 469435978, 744740232

Offset: 1

Alonso del Arte, Mar 17 2011

Among bases b = 2 to 36, in b = 5 there is the third highest percentage of Keith numbers between b and b^3 (that is, numbers with two or three digits); only binary and ternary have more Keith numbers in that range.

			a(2) = 9. In base 5, the number 9 is written 14, and the second order linear recurrence is then 1, 4, 5, 9, ... therefore 9 is a Keith number in base 5.
The number 14 is a Keith number in base 10 but not base 5, as we have: 2, 4, 6, 10, 16, ...

Cf. A007629, base-10 Keith numbers; A162724, binary Keith numbers.

(* First run the program for A186830 to define keithSeq *) Select[Range[5, 10^6], Last[keithSeq[#, 5]] == # &]

Extended by T. D. Noe, Mar 23 2011

A188195 Base-3 Keith numbers.

Original entry on oeis.org

3, 5, 6, 7, 57, 102, 127, 206, 217, 677, 805, 840, 1486, 1680, 1887, 2090, 2834, 8329, 10145, 12866, 21127, 23002, 50782, 67925, 82685, 96841, 153861, 178852, 357896, 3826652, 17985694, 38610616, 38610808, 70587766, 160804168, 341014432, 632582224

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629.

			57 is here because, in base 3, 57 is 2010 and applying the Keith iteration to this number produces the numbers 2, 0, 1, 0, 3, 4, 8, 15, 30, 57.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188196-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,3]&]

A188200 Base-9 Keith numbers.

Original entry on oeis.org

17, 21, 25, 42, 67, 81, 96, 101, 149, 162, 173, 202, 243, 303, 324, 346, 404, 405, 486, 519, 567, 648, 692, 732, 857, 1189, 1464, 2199, 4398, 11644, 18325, 33774, 34453, 37999, 70348, 92664, 141557, 256820, 263412, 326778, 349484, 526824, 535754, 579708, 1461987, 1519308, 1621052, 2688905, 4650964, 8027458, 8198651, 8374956, 13504910, 17858551, 20002383, 55640285, 154513633, 170801638

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629. Base 9 appears to be unusually rich in Keith numbers. Why?

			101 is here because, in base 9, 101 is 122 and applying the Keith iteration to this number produces the numbers 1, 2, 2, 5, 9, 16, 30, 55, 101. Note that the multiples 202, 303, and 404 are here also.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188199.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,9]&]

A188201 The least base-n Keith number >= n.

Original entry on oeis.org

2, 3, 5, 5, 8, 8, 8, 17, 14, 13, 13, 13, 20, 18, 23, 33, 26, 21, 21, 21, 32, 28, 35, 49, 29, 33, 41, 57, 44, 38, 34, 34, 34, 43, 53, 73, 56, 48, 45, 81, 62, 53, 47, 89, 68, 53, 71, 97, 74, 63, 77, 55, 55, 55, 60, 113, 86, 73, 89, 69, 92, 78, 95, 129, 98, 83, 73, 137, 104, 88

Offset: 2

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629. It appears that a(n) < 2n. If n or n+1 is a Fibonacci number f, then a(n) = f. If n>3 and n+2 is a Fibonacci number f, then a(n) = f. The graph shows that 2n-1, 3n/2-1, and 8(n-5)/7+5 are frequent values of a(n).

T. D. Noe, Table of n, a(n) for n = 2..1000

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Table[k = n; While[! IsKeith[k, n], k++]; k, {n, 2, 100}]

A188381 Negabinary Keith numbers.

Original entry on oeis.org

2, 3, 4, 7, 9, 13, 16, 36, 55, 64, 162, 256, 458, 1024, 1829, 4096, 7316, 15119, 16384, 18970, 37702, 37723, 45171, 60476, 65536, 84506, 262144, 277263, 1048576, 1109052, 1722002, 2160570, 4194304, 10549178, 12699958, 15084573, 16777216, 31921069, 67108864

Offset: 1

Alonso del Arte, Mar 29 2011

Keith numbers are described in A007629. All powers of 4 appear. However, 2 is the only number of the form 2^n with n odd that appears in the sequence. That's because in negabinary, such numbers are represented as 11 followed by n 0's, and that leads to the sequence 1, 1, 0, ... , 0, 2, 3, 5, 10, 20, 40, 80, 160, ... up to 5(2^(n - 2)), and 5(2^(n - 2)) > 2^(n - 1). (See A020714).

Cf. A007629, A039724, A162724.

(* First run the program from A039724 to define ToNegaBases *) keithFromListQ[n_Integer, digits_List] := Module[{seq = digits, curr = digits[[-1]], ord = Length[digits]}, While[curr < n, curr = Plus@@Take[seq, -ord]; AppendTo[seq, curr]]; Return[seq[[-1]] == n]]; Select[Range[2, 32768], keithFromListQ[#, IntegerDigits[ToNegaBases[#, 2]]] &]

a(33)-a(39) from Amiram Eldar, Jan 29 2020

A188196 Base-4 Keith numbers.

Original entry on oeis.org

5, 7, 10, 15, 18, 29, 47, 113, 163, 269, 1150, 1293, 1881, 22173, 44563, 95683, 261955, 1179415, 1295936, 11451171, 26867679, 42531919, 247791599, 429914163, 445379527, 560533869, 619222313, 2147478019, 2971786617, 3474640372

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629.

			47 is here because, in base 4, 47 is 233 and applying the Keith iteration to this number produces the numbers 2, 3, 3, 8, 14, 25, 47.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,4]&]

A188199 Base-8 Keith numbers.

Original entry on oeis.org

8, 11, 15, 16, 22, 24, 32, 37, 40, 48, 56, 59, 92, 123, 200, 251, 257, 400, 457, 893, 2761, 4040, 4547, 8392, 9161, 12833, 16784, 21225, 29617, 127126, 238244, 378733, 526117, 587524, 599333, 672549, 745765, 2176234, 2347267, 2593739, 5285583, 8113400, 341785390, 449415500, 514971408

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629.

			200 is here because, in base 8, 200 is 310 and applying the Keith iteration to this number produces the numbers 3, 1, 0, 4, 5, 9, 18, 32, 59, 109, 200.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,8]&]

A188197 Base-6 Keith numbers.

Original entry on oeis.org

8, 11, 16, 27, 37, 44, 74, 88, 111, 148, 185, 409, 526, 2417, 8720, 12154, 15268, 49322, 61587, 68444, 82833, 98644, 206356, 249549, 327001, 484512, 642437, 692928, 695659, 726975, 964225, 1210087, 2141228, 2282504, 5514048, 10640601, 48453362, 69572128, 74343984, 171550728, 184847569, 204545417, 232877871, 245317977, 246133682

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629.

			44 is here because, in base 6, 44 is 112 and applying the Keith iteration to this number produces the numbers 1, 1, 2, 4, 7, 13, 24, 44.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,6]&]

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A265426 Primes p such that p - 1 is a binary Keith number (A162724).

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

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A187713 Base-5 Keith numbers.

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A188195 Base-3 Keith numbers.

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A188200 Base-9 Keith numbers.

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A188201 The least base-n Keith number >= n.

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A188381 Negabinary Keith numbers.

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A188196 Base-4 Keith numbers.

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