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.

Showing 1-3 of 3 results.

A100401 Digital root of 3^n.

Original entry on oeis.org

1, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 0

Views

Author

Cino Hilliard, Dec 30 2004

Keywords

Comments

This sequence also gives the digital root of 12^n, 21^n, 30^n, 39^n, 48^n, 57^n, ... (any k^n where k is congruent to 3 mod 9). - Timothy L. Tiffin, Dec 02 2023

Examples

			For n=14, the digits of 3^14 = 4782969 sum to 45, whose digits sum to 9. So, a(14) = 9.

Links

Crossrefs

Cf. A000244, A001021, A007953, A009965, A009974, A009983, A009992, A010888, A100403, A225374.

Programs

Formula

a(n) = 3^n mod 18. - Zerinvary Lajos, Nov 25 2009
From Timothy L. Tiffin, Nov 30 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+2x+6x^2)/(1-x).
a(n) = A100403(n) for n <> 1. (End)
a(n) = A010888(A000244(n)). - Michel Marcus, Dec 01 2023
a(n) = A010888(A001021(n)) = A010888(A009965(n)) = A010888(A009974(n)) = A010888(A009983(n)) = A010888(A009992(n)) = A010888(A225374(n)). - Timothy L. Tiffin, Dec 02 2023
E.g.f.: 9*exp(x) - 6*x - 8. - Elmo R. Oliveira, Aug 08 2024
a(n) = A007953(3*a(n-1)) = A010888(3*a(n-1)). - Stefano Spezia, Mar 20 2025

A381487 Numbers which are a power of their digital root.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 128, 256, 512, 729, 2401, 6561, 8192, 16384, 32768, 59049, 78125, 524288, 531441, 823543, 1048576, 2097152, 4782969, 33554432, 43046721, 67108864, 134217728, 282475249, 387420489, 1220703125, 2147483648, 3486784401, 4294967296
Offset: 1

Views

Author

Stefano Spezia, Feb 25 2025

Keywords

Examples

			a(12) = 128 is a term since 128 = 2^7 = A010888(128)^7.

Links

Crossrefs

Cf. A010888, A023106, A381491, A381492.
Digital root of k^n: A000012 (1), A153130 (2), A100401 (3), A100402 (4), A070366 (5), A100403 (6), A070403 (7), A010689 (8), A010734 (9).

Programs

  • Mathematica
    A010888[n_]:=n - 9*Floor[(n-1)/9]; kmax=5*10^6; a={0,1}; For[k=2, k<=kmax, k++, If[A010888[k]!=1, If[IntegerQ[Log[A010888[k],k]], AppendTo[a,k]]]]; a
  • PARI
    isok(k) = if ((k==0) || (k==1), return(1)); my(d=(k-1)%9+1); if (d>1, d^logint(k, d) == k); \\ Michel Marcus, Feb 26 2025
  • PARI
    lista(nn) = my(list = List()); listput(list, 0); listput(list, 1); for (n=2, 9, for (k=1, logint(nn, n), if ((n^k-1)%9+1 == n, listput(list, n^k)););); vecsort(Vec(list)); \\ Michel Marcus, Feb 27 2025

Formula

a(n) = A381491(n)^A381492(n).

A291092 1 followed by infinitely many 9's.

Original entry on oeis.org

1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 1

Views

Author

N. J. A. Sloane, Aug 19 2017

Keywords

Comments

The digital root of 9^(n-1). - Cino Hilliard, Dec 31 2004
With interpolated zeros (1,0,9,0,9,0,9,0,...) this is the number of hours between times when the hands of a two-handed clock cross. - Halfdan Skjerning, Aug 18 2017

Links

Crossrefs

Cf. A178181, A010734, A180599, A100403, A100401.

Programs

  • Mathematica
    PadRight[{1},100,9] (* Paolo Xausa, Oct 16 2023 *)

Formula

G.f.: x*(1 + 8*x)/(1 - x). - Chai Wah Wu, Aug 19 2017
E.g.f.: 9*(exp(x) - 1) - 8*x. - Stefano Spezia, Oct 16 2023
Showing 1-3 of 3 results.

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

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