A013733 -id:A013733 - cp's OEIS frontend

A013732 a(n) = 3^(3*n + 1).

3, 81, 2187, 59049, 1594323, 43046721, 1162261467, 31381059609, 847288609443, 22876792454961, 617673396283947, 16677181699666569, 450283905890997363, 12157665459056928801, 328256967394537077627, 8862938119652501095929

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (27).

Cf. A013730, A013733, A013734.

[3^(3*n+1): n in [0..25]]; // Vincenzo Librandi, May 25 2011

seq(3^(3*n+1),n=0..15); # Nathaniel Johnston, Jun 26 2011

NestList[27#&,3,20] (* Harvey P. Dale, Apr 01 2018 *)

a(n)=3^(3*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 27*a(n-1); a(0)=3.
G.f.: 3/(1-27*x). (End)

A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

Original entry on oeis.org

9, 16, 128, 243, 576, 1024, 6561, 8192, 9604, 11664, 28224, 36864, 51984, 65536, 97344, 140625, 177147, 250000, 275625, 345744, 419904, 450241, 524288, 614656, 717409, 746496, 1028196, 1058841, 1399489, 1500625, 1590121, 1750329, 1806336, 1882384, 2359296, 3326976, 4194304

Offset: 1

Altug Alkan, Jan 10 2016

nonn

Intersection of A001597 and A003325.
Motivation for this sequence is the equation m^k = x^3 + y^3 where x,y,m > 0 and k >= 2.
Obviously, because of Fermat's Last Theorem, a(n) cannot be a cube.
A050802 is a subsequence.
Obviously, this sequence contains all numbers of the form 2^(3*n+1) and 3^(3*n-1), for n > 0.

			9 is a term because 9 = 3^2 = 1^3 + 2^3.
16 is a term because 16 = 2^4 = 2^3 + 2^3.
243 is a term because 243 = 3^5 = 3^3 + 6^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5012

Cf. A001597, A003325, A050802, A085332, A013730, A013733, A266927.

T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=2, 1e7, if(ispower(n) && is(n), print1(n, ", ")))

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

Paul Curtz, Jun 02 2016

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).

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

Table[Denominator[n/3^n], {n, 0, 28}] (* Michael De Vlieger, Jun 03 2016 *)

a(n) = denominator(n/3^n) \\ Felix Fröhlich, Jun 07 2016

[1] + [3^(n-n.valuation(3)) for n in [1..30]] # Tom Edgar, Jun 02 2016

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

Henry Bottomley, Jun 20 2000

Index entries for linear recurrences with constant coefficients, signature (0, 27).

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.

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 *)

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).

A383348 Triangle related to the partitions of n in three colors, read by rows.

Original entry on oeis.org

9, 6, 243, 1, 243, 6561, 0, 90, 8748, 177147, 0, 15, 4860, 295245, 4782969, 0, 1, 1458, 216513, 9565938, 129140163, 0, 0, 252, 91854, 8680203, 301327047, 3486784401, 0, 0, 24, 24786, 4723920, 325241892, 9298091736, 94143178827, 0, 0, 1, 4374, 1712421, 215233605, 11622614670, 282429536481, 2541865828329

Offset: 1

Michel Marcus, Apr 24 2025

			Triangle begins:
  9;
  6, 243;
  1, 243, 6561;
  0, 90, 8748, 177147;
  0, 15, 4860, 295245, 4782969;
  ...

D. S. Gireesh and M. S. Mahadeva Naika, On 3-regular partitions in 3-colors, Indian J. Pure Appl. Math. 50 (2019), 137-148.

D. S. Gireesh and M. S. Mahadeva Naika, On 3-regular partitions in 3-colors, ResearchGate.
B. Hemanthkumar and D. S. Gireesh, On ℓ-regular and 2-color partition triples modulo powers of 3, arXiv:2504.13507 [math.CO], 2025.

Cf. A013733 (diagonal).

M(i,j) = if (j>i, return(0)); if (i==1, if (j==1, return(9))); if (i==2, if (j==1, return(6)); return(243)); if (i==3, if (j==1, return(1)); if (j==2, return(243)); return(6561)); if (i>=4, if (j==1, return(0)); 27*M(i-1,j-1) + 9*M(i-2,j-1) + M(i-3,j-1));
row(n) = vector(n, i, M(n, i));

T(i,j) = 27*T(i-1,j-1) + 9*T(i-2,j-1) + T(i-3,j-1).

cp's OEIS Frontend

A013732 a(n) = 3^(3*n + 1).

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A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

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A273893 Denominator of n/3^n.

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

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A383348 Triangle related to the partitions of n in three colors, read by rows.

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