cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-50 of 63 results. Next

A282762 7*n analog to Keith numbers.

Original entry on oeis.org

3, 6, 9, 12, 25, 29, 33, 58, 62, 66, 70, 87, 91, 95, 99, 124, 128, 150, 152, 165, 178, 191, 204, 217, 592, 801, 1184, 3860, 15728, 59800, 117711, 157701, 230720, 270737, 496085, 795918, 869366, 954639, 1549319, 1826773, 3169440, 3170466, 3973793, 3974819, 3975845, 4012718, 4013744, 5120160, 5653357, 5978943
Offset: 1

Views

Author

Paolo P. Lava, Feb 22 2017

Keywords

Comments

Like Keith numbers but starting from 7*n digits to reach n.
Consider the digits of 7*n. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Examples

			7*25 = 175:
1 + 7 + 5 = 13;
7 + 5 + 13 =25.

Crossrefs

Cf. A282757 - A282761, A282763 - A282765.

Programs

  • Maple
    with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]
  • Mathematica
    Select[Range[10^6], Function[n, Module[{d = IntegerDigits[7 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)

A282763 8*n analog to Keith numbers.

Original entry on oeis.org

9, 20, 176, 184, 277, 2669, 15705, 233202, 241202, 445657, 742714, 2095479, 4697536, 10508788, 20308656, 55683878, 86603874
Offset: 1

Views

Author

Paolo P. Lava, Feb 22 2017

Keywords

Comments

Like Keith numbers but starting from 8*n digits to reach n.
Consider the digits of 8*n. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Examples

			8*20 = 160:
1 + 6 + 0 = 7;
6 + 0 + 7 = 13;
0 + 7 + 13 = 20.

Crossrefs

Cf. A282757 - A282762, A282764, A282765.

Programs

  • Maple
    with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]
  • Mathematica
    Select[Range[10^6], Function[n, Module[{d = IntegerDigits[8 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)

A282764 9*n analog to Keith numbers.

Original entry on oeis.org

9, 17, 48, 55, 96, 120, 124, 131, 244, 426, 787, 1893, 5307, 5364, 5600, 10083, 31085, 46733, 52700, 53456, 56857, 56920, 109620, 110313, 110376, 374016, 2989245, 4081505, 5173765, 13017112, 17242512, 34346372, 34638676
Offset: 1

Views

Author

Paolo P. Lava, Feb 22 2017

Keywords

Comments

Like Keith numbers but starting from 9*n digits to reach n.
Consider the digits of 9*n. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Examples

			9*17 = 153:
1 + 5 + 3 = 9;
5 + 3 + 9 = 17.

Crossrefs

Cf. A282757 - A282763, A282765.

Programs

  • Maple
    with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]
  • Mathematica
    Select[Range[10^6], Function[n, Module[{d = IntegerDigits[9 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)

A377416 Least integer k which, by a process analogous to the Keith numbers, reach k + n.

Original entry on oeis.org

14, 12, 11, 10, 16, 13, 17, 37, 18, 12, 11, 10, 40, 15, 27, 39, 13, 16, 24, 67, 22, 17, 12, 11, 10, 18, 21, 36, 43, 19, 15, 58, 23, 30, 13, 51, 48, 16, 54, 27, 44, 38, 34, 12, 11, 10, 14, 91, 20, 32, 55, 18, 42, 29, 35, 21, 25, 277, 15, 150, 66, 72, 56, 13, 46
Offset: 0

Views

Author

Paolo P. Lava, Oct 27 2024

Keywords

Examples

			a(5) = 13 because 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18 that is 13 + 5.

Links

Crossrefs

Cf. A007629, A377439.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,c,j,k,n,t,v; v:=array(1..h); c:=[];
for j from 0 to 59 do for n from 10 to q do a:=n; b:=length(a);
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k],k=1..b);
while v[t]

A006576 Primitive repfigit numbers.

Original entry on oeis.org

14, 19, 47, 61, 75, 197, 742, 1104, 1537, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 86935, 120284, 129106, 147640, 156146, 174680, 183186, 298320
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Keith, Repfigit Numbers, J. Recreational Math., Vol. 19, No. 2, pp. 41-42, 1987.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A007629.

Extensions

a(19) corrected by Sean A. Irvine, May 12 2017

A048970 Prime Keith numbers.

Original entry on oeis.org

19, 47, 61, 197, 1084051, 74596893730427
Offset: 1

Views

Author

G. L. Honaker, Jr.

Keywords

Links

Crossrefs

Cf. A007629.

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

Views

Author

T. D. Noe, Mar 24 2011

Keywords

Comments

Keith numbers are described in A007629.

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

T. D. Noe, Mar 24 2011

Keywords

Comments

Keith numbers are described in A007629.

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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]&]

A243021 Numbers n such that n appears in the sequence beginning with the digit-sum of n and extended by adding successive digit-sums.

Original entry on oeis.org

12, 15, 18, 21, 24, 27, 30, 32, 33, 36, 37, 39, 45, 47, 51, 52, 54, 57, 58, 59, 62, 63, 69, 72, 81, 84, 90, 96, 99, 101, 103, 107, 109, 111, 114, 115, 117, 119, 120, 123, 126, 129, 135, 141, 144, 147, 153, 159, 161, 162, 163, 169, 171, 173, 174, 180, 184, 185, 186, 189, 197, 199, 201, 204, 207, 210, 213, 216, 219, 221, 225, 226, 231, 234, 236, 237
Offset: 1

Views

Author

Anthony Sand, May 29 2014

Keywords

Comments

Numbers n>9 with following property: form a sequence b(i) whose initial term is digit-sum(n). Later terms are given by the rule that b(i) = b(i-1) + digit-sum(b(i-1)) and n itself appears in the sequence.

Examples

			When n = 0..9, the sequence immediately produces n. The sequence for 12 is seeded with digital-sum(12) = 3, 3 + 3 = 6, 6 + 6 = 12. 15 yields 6, 6 + 6 = 12, 12 + 3 = 15. 18 yields 9, 9 + 9 = 18. 21 yields 3, 3 + 3 = 6, 6 + 6 = 12, 12 + 3 = 15, 15 + 6 = 21.

Crossrefs

Cf. A007629, A243140.

Formula

b(i) = b(i-1) + digit-sum(b(i-1))

Extensions

Added comments based on those for A007629.

A263535 a(1) = 1; thereafter a(n) = a(n-1) + d_1^1 + d_2^2 + d_3^3 + ..., where d_1 d_2 d_3 ... is the decimal expansion of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 53, 67, 122, 135, 270, 321, 329, 1065, 1907, 4390, 5132, 5181, 5700, 5754, 6189, 13269, 73632, 73977, 93930, 94758, 128519, 661103, 661876, 729478, 1009425, 1095200, 1096587, 2187425, 2269554, 2311471, 2430158, 4542981, 4864284, 5143384, 5422306
Offset: 1

Views

Author

Pieter Post, Oct 20 2015

Keywords

Comments

This additive sequence will tend to be geometric.

Examples

			a(5)=16, so a(6) is 16 + 1^1 + 6^2 = 53.

Links

Crossrefs

Cf. A007629, A005188.

Programs

  • Mathematica
    NestList[#+Total[IntegerDigits[#]^Range[IntegerLength[#]]]&,1,40] (* Harvey P. Dale, Jan 19 2021 *)
  • PARI
    lista(nn) = {print1(a=1, ", "); for (n=2, nn, d = digits(a); na = a + sum(i=1, #d, d[i]^i); print1(na, ", "); a = na;);} \\ Michel Marcus, Nov 20 2015
  • Python
    def moda(n):
    return sum(int(d)**(i + 1) for i, d in enumerate(str(n)))
b = 1
resu = [1]
for a in range(1, 100):
    b += moda(b)
    resu.append(b)
resu
  • Sage
    A=[1]
for i in [1..2000]:
    A.append(A[i-1]+sum(A[i-1].digits()[len(A[i-1].digits())-1-j]^(j+1) for j in [0..len(A[i-1].digits())-1]))
A # Tom Edgar, Oct 20 2015
Previous Showing 41-50 of 63 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.