A281917 -id:A281917 - cp's OEIS frontend

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

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

A281915 4th power analog of Keith numbers.

Original entry on oeis.org

1, 7, 19, 20, 22, 25, 28, 36, 77, 107, 110, 175, 789, 1528, 1932, 3778, 5200, 7043, 8077, 38855, 41234, 44884, 49468, 204386, 763283, 9423515, 73628992, 87146144, 146124072, 146293356, 326194628, 1262293219, 1321594778, 2767787511, 11511913540, 12481298961, 13639550655

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^4 digits to reach n.

Consider the digits of n^4. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			175^4 = 937890625:
9 + 3 + 7 + 8 + 9 + 0 + 6 + 2 + 5 = 49;
3 + 7 + 8 + 9 + 0 + 6 + 2 + 5 + 49 = 89;
7 + 8 + 9 + 0 + 6 + 2 + 5 + 49 + 89 = 175.

Cf. A055575, A007629, A246544, A263534.

Cf. A274769, A274770, A281916, A281917, A281918, A281919, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281915[n_] := Join[{1, 7}, Select[Range[10, n], keithQ[#, 4]&]]
a281915[10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

a(27)-a(28) from Jinyuan Wang, Jan 30 2020

Missing a(25) and a(29)-a(37) from Giovanni Resta, Jan 31 2020

A281916 5th power analog of Keith numbers.

Original entry on oeis.org

1, 28, 35, 36, 46, 51, 99, 109, 191, 239, 476, 491, 1022, 1126, 1358, 1362, 15156, 21581, 44270, 63377, 100164, 375830, 388148, 2749998, 5215505, 10158487, 81082532, 87643314, 410989134, 1485204944, 3496111364, 3829840893, 15889549579, 16107462404, 16766005098, 17608009898

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^5 digits to reach n.

Consider the digits of n^5. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			109^5 = 15386239549:
1 + 5 + 3 + 8 + 6 + 2 + 3 + 9 + 5 + 4 + 9 = 55;
5 + 3 + 8 + 6 + 2 + 3 + 9 + 5 + 4 + 9 + 55 = 109.

Cf. A055576, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281917, A281918, A281919, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281916[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 5]&]]
a281916[5*10^5] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(27)-a(28) from Jinyuan Wang, Jan 31 2020

a(29)-a(36) from Giovanni Resta, Jan 31 2020

A281918 7th power analog of Keith numbers.

Original entry on oeis.org

1, 18, 27, 31, 34, 43, 53, 58, 68, 145, 187, 314, 826, 2975, 37164, 40853, 58530, 72795, 77058, 160703, 187617, 1926759, 6291322, 6628695, 25285305, 31292514, 33968486, 54954185, 71593237, 125921697, 555963577, 575307142, 2393596216, 2444508547, 42544333760, 97812197525

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^7 digits to reach n.
Consider the digits of n^7. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.
a(30) > 10^8. - Jinyuan Wang, Jan 30 2020

			145^7 = 1347646586640625:
1 + 3 + 4 + 7 + 6 + 4 + 6 + 5 + 8 + 6 + 6 + 4 + 0 + 6 + 2 + 5 = 73;
3 + 4 + 7 + 6 + 4 + 6 + 5 + 8 + 6 + 6 + 4 + 0 + 6 + 2 + 5 + 73 = 145.

Cf. A226971, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281919, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281918[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 7]&]]
a281918[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(28)-a(29) from Jinyuan Wang, Jan 30 2020

a(30)-a(36) from Giovanni Resta, Feb 03 2020

A281919 8th-power analog of Keith numbers.

Original entry on oeis.org

1, 30, 46, 54, 63, 207, 394, 693, 694, 718, 20196, 42664, 80051, 90135, 91447, 93136, 207846, 324121, 361401, 421609, 797607, 802702, 882227, 1531788, 1788757, 1789643, 4028916, 4176711, 6692664, 15643794, 31794346, 65335545, 140005632, 144311385, 153364253

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^8 digits to reach n.

Consider the digits of n^8. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

Giovanni Resta, Table of n, a(n) for n = 1..40

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281919[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 8]&]]
a281919[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

207^8 = 3371031134626313601:
+ 3 + 7 + 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 = 54;
+ 7 + 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 + 54 = 105;
+ 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 + 54 + 105 = 207.

a(32) from Jinyuan Wang, Feb 01 2020

Terms a(33) and beyond from Giovanni Resta, Feb 03 2020

A281920 9th-power analog of Keith numbers.

Original entry on oeis.org

1, 54, 71, 81, 196, 424, 451, 2394, 9057, 51737, 52141, 104439, 227914, 228088, 1019555, 1096369, 1202713, 1687563, 1954556, 3332130, 3652731, 4177592, 31669012, 79937731, 81478913, 148341053, 168763202, 182573136, 342393476, 773367191, 1450679282, 2914657310, 3282344153

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^9 digits to reach n.

Consider the digits of n^9. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			196^9 = 426878854210636742656:
4 + 2 + 6 + 8 + 7 + 8 + 8 + 5 + 4 + 2 + 1 + 0 + 6 + 3 + 6 + 7 + 4 + 2 + 6 + 5 + 6 = 100;
2 + 6 + 8 + 7 + 8 + 8 + 5 + 4 + 2 + 1 + 0 + 6 + 3 + 6 + 7 + 4 + 2 + 6 + 5 + 6 + 100 = 196.

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281920[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 9]&]]
a281920[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(24) from Jinyuan Wang, Feb 02 2020

a(25)-a(33) from Giovanni Resta, Feb 03 2020

A281921 10th-power analog of Keith numbers.

Original entry on oeis.org

1, 82, 85, 94, 97, 106, 117, 459, 1197, 24615, 24657, 26184, 87664, 117099, 538168, 1049708, 1229174, 2210323, 4587773, 11019224, 96167938, 104719358, 202511251, 226456915, 821871524, 1811437987, 1832881095, 3530066559, 7414362499, 7906250753, 15607432165, 15631766564

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^10 digits to reach n.

Consider the digits of n^10. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			106^10 = 179084769654285362176: 1 + 7 + 9 + 0 + 8 + 4 + 7 + 6 + 9 + 6 + 5 + 4 + 2 + 8 + 5 + 3 + 6 + 2 + 1 + 7 + 6 = 106.

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281921[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 10]&]]
a281921[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(21) from Jinyuan Wang, Feb 02 2020

a(22)-a(32) from Giovanni Resta, Feb 03 2020

A346067 Smallest prime that is the n-th power analog of Keith numbers.

Original entry on oeis.org

2, 37, 17, 7, 109, 36013476739, 31, 80051, 71, 97, 107, 13093, 103, 127, 107, 163, 991, 181, 157, 181, 199, 193, 271, 31663, 211, 307, 307, 5318989651, 673, 8297, 331, 811, 359, 463

Offset: 1

Giorgos Kalogeropoulos, Jul 03 2021

The n-th power analog of Keith numbers is like Keith numbers but starting from p^n to reach p. Consider the digits of p^n where p is prime. Take their sum and repeat the process, deleting the first addend and adding the previous sum. We are searching for the first prime p that after some number of iterations reaches a sum equal to p.
The only terms for n <= 100 whose values are still unknown are a(35), a(90), a(91) and a(95).
Paolo Lava asked for these numbers as a puzzle (see the Rivera link) and as a result a(61) = 11659149703 and a(81) = 200908021 were found.

			a(2) = 37 because 37^2 = 1369. Then 1+3+6+9 = 19 and 3+6+9+19 = 37.

Carlos Rivera, Puzzle 1043. Another puzzle about Keith numbers, The Prime Puzzles & Problems Connection.

Cf. A007629 (Keith numbers).

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

KeithPowQ[m_Integer,n_]:=Module[{b=IntegerDigits[m^n],s,k=0},s=Total[b];While[sA007629 *)

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

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A281915 4th power analog of Keith numbers.

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A281916 5th power analog of Keith numbers.

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A281918 7th power analog of Keith numbers.

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A281919 8th-power analog of Keith numbers.

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A281920 9th-power analog of Keith numbers.

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A281921 10th-power analog of Keith numbers.

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