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.

A013733 a(n) = 3^(3n+2).

Original entry on oeis.org

9, 243, 6561, 177147, 4782969, 129140163, 3486784401, 94143178827, 2541865828329, 68630377364883, 1853020188851841, 50031545098999707, 1350851717672992089, 36472996377170786403, 984770902183611232881, 26588814358957503287787
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Additive digital root of a(n) is equal to 9. - Miquel Cerda, Oct 31 2016

Links

Crossrefs

Cf. A013731, A013732, A013735, A013737.

Programs

Formula

a(n)=27*a(n-1), n>0 ; a(0)=9 . G.f.: 9/(1-27*x). - Philippe Deléham, Nov 25 2008

A273893 Denominator of n/3^n.

Original entry on oeis.org

1, 3, 9, 9, 81, 243, 243, 2187, 6561, 2187, 59049, 177147, 177147, 1594323, 4782969, 4782969, 43046721, 129140163, 43046721, 1162261467, 3486784401, 3486784401, 31381059609, 94143178827, 94143178827, 847288609443, 2541865828329, 282429536481, 22876792454961
Offset: 0

Views

Author

Paul Curtz, Jun 02 2016

Keywords

Comments

The reduced values are Ms(n) = 0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, 7/2187, 8/6561, 1/2187, ... .
Numerators: 0, 1, 2, 1, 4, ... = A038502(n).
Ms(-n) = 0, -3, -18, ... = - A036290(n).
Difference table of Ms(n):
0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, ...
1/3, -1/9, -1/9, -5/81, -7/243, -1/81, ...
-4/9, 0, 4/81, 8/243, 4/243, ...
4/9, 4/81, -4/243, -4/243, ...
-32/81, -16/243, 0, ...
80/243, 16/243, ...
-64/243, ...
etc.
The difference table of O(n) = n/2^n (Oresme numbers) has its 0's on the main diagonal. Here the 0's appear every two rows. For n/4^n,they appear every three rows. (The denominators of O(n) are 2^A093048(n)).
All terms are powers of 3 (A000244).

Crossrefs

Cf. A007949, A000244, A013732, A013733, A036290, A038502, A075101.

Programs

  • Mathematica
    Table[Denominator[n/3^n], {n, 0, 28}] (* Michael De Vlieger, Jun 03 2016 *)
  • PARI
    a(n) = denominator(n/3^n) \\ Felix Fröhlich, Jun 07 2016
  • Sage
    [1] + [3^(n-n.valuation(3)) for n in [1..30]] # Tom Edgar, Jun 02 2016

Formula

For n>0, a(n) = 3^(n - valuation(n,3)) = 3^(n - A007949(n)). - Tom Edgar, Jun 02 2016
a(3n+1) = 3^(3n+1), a(3n+2) = 3^(3n+2).
a(3n+6) = 27*(3n+3).
From Peter Bala, Feb 25 2019: (Start)
a(n) = 3^n/gcd(n,3^n).
O.g.f.: 1 + F(3*x) - (2/3)*F((3*x)^3) - (2/9)*F((3*x)^9) - (2/27)*F((3*x)^27) - ..., where F(x) = x/(1 - x).
O.g.f. for reciprocals: Sum_{n >= 0} x^n/a(n) = 1 + F((x/3)) + 2*( F((x/3)^3) + 3*F((x/3)^9) + 9*F((x/3)^27) + ... ). Cf. A038502. (End)

A055156 Powers of 3 which are not powers of 3^3.

Original entry on oeis.org

3, 9, 81, 243, 2187, 6561, 59049, 177147, 1594323, 4782969, 43046721, 129140163, 1162261467, 3486784401, 31381059609, 94143178827, 847288609443, 2541865828329, 22876792454961, 68630377364883, 617673396283947
Offset: 0

Views

Author

Henry Bottomley, Jun 20 2000

Keywords

Links

Crossrefs

Cf. A013732 and A013733. Consists of numbers in A000244 which are not in A009971. See A004171 for powers of 2 which are not powers of 2^2.

Programs

  • Mathematica
    With[{nn=40},Complement[3^Range[nn],27^Range[Floor[nn/3]]]] (* or *) LinearRecurrence[{0,27},{3,9},40] (* Harvey P. Dale, Jul 17 2012 *)

Formula

a(n) = a(n-1)*a(n-2)/a(n-3) = 27*a(n-2) = 3^A001651(n).
a(2n) = 3^(3n+1), a(2n+1) = 3^(3n+2).
Showing 1-3 of 3 results.

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

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