A092763 -id:A092763 - cp's OEIS frontend

A138256 Smallest positive integer m such that n divides [3^m/m] (=A092763(m)).

1, 2, 1, 2, 4, 5, 13, 5, 3, 4, 6, 5, 7, 13, 13, 5, 16, 10, 15, 4, 13, 11, 15, 5, 22, 7, 9, 13, 30, 13, 33, 17, 11, 16, 13, 10, 19, 37, 7, 13, 8, 13, 55, 11, 14, 67, 35, 5, 43, 22, 82, 7, 108, 92, 20, 13, 37, 30, 120, 13, 11, 33, 26, 17, 14, 11, 23, 16, 24, 13, 72, 10, 13, 19, 22, 37

Offset: 1

Max Alekseyev, Mar 09 2008

nonn

This sequence is well-defined.

Romanian Master in Mathematics Contest, Bucharest, 2007, Problem 3

Cf. A092763, A138255, A138257, A138258, A138259, A138260, A138261, A138262, A138263.

pim[n_]:=Module[{m=1},While[!Divisible[Floor[3^m/m],n],m++];m]; Array[pim, 80] (* Harvey P. Dale, Oct 26 2011 *)

A129795 a(n) = floor(5^n/n).

Original entry on oeis.org

5, 12, 41, 156, 625, 2604, 11160, 48828, 217013, 976562, 4438920, 20345052, 93900240, 435965401, 2034505208, 9536743164, 44878791360, 211927625868, 1003867701480, 4768371582031, 22706531343005, 108372081409801

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(5^n / n): n in [1..35]]; // Vincenzo Librandi, May 08 2011

Table[Floor[5^n/n],{n,30}] (* Harvey P. Dale, Jul 19 2011 *)

A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 4, 9, 8, 5, 0, 6, 20, 21, 12, 6, 0, 10, 48, 64, 41, 18, 7, 0, 18, 121, 204, 156, 72, 24, 8, 0, 32, 312, 682, 625, 324, 114, 32, 9, 0, 56, 820, 2340, 2604, 1555, 600, 170, 40, 10, 0, 102, 2187, 8192, 11160, 7776, 3361, 1024, 243, 50, 11

Offset: 1

Henry Bottomley, Mar 12 2001

			T(5,3)=[5^3/3]=[125/3]=41.
Rows start:
  1,  0,  0,   0,   0, ...
  2,  2,  2,   4,   6, ...
  3,  4,  9,  20,  48, ...
  4,  8, 21,  64, 204, ...
  5, 12, 41, 156, 625, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Rows include A000007, A000799, A092763, A129794, A129795, A129796, A129797, A129798, A129799, A060156.
Columns include A000027, A007590.
Diagonals include A000169.

T(n, k) = (A051129(n, k)-A060154(n, k))/k.

A129794 a(n) = floor(4^n/n).

Original entry on oeis.org

4, 8, 21, 64, 204, 682, 2340, 8192, 29127, 104857, 381300, 1398101, 5162220, 19173961, 71582788, 268435456, 1010580540, 3817748707, 14467258260, 54975581388, 209430786243, 799644820200, 3059510616420, 11728124029610

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(4^n / n): n in [1..35]]; // Vincenzo Librandi, May 08 2011

Table[Floor[4^n/n],{n,25}] (* Harvey P. Dale, Jul 24 2011 *)

def A129794(n): return (1<<(n<<1))//n # Chai Wah Wu, Aug 24 2023

A129796 a(n) = floor(6^n/n).

Original entry on oeis.org

6, 18, 72, 324, 1555, 7776, 39990, 209952, 1119744, 6046617, 32981550, 181398528, 1004668770, 5597440292, 31345665638, 176319369216, 995685849690, 5642219814912, 32071565263710, 182807922003148, 1044616697160850

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(6^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

Table[Floor[6^n/n],{n,30}] (* Harvey P. Dale, Feb 20 2015 *)

A129797 a(n) = floor(7^n/n).

Original entry on oeis.org

7, 24, 114, 600, 3361, 19608, 117649, 720600, 4483734, 28247524, 179756976, 1153440600, 7453000800, 48444505203, 316504100662, 2077058160600, 13684147881600, 90467422106136, 599941851861744, 3989613314880600

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(7^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

Table[Floor[7^n/n],{n,30}] (* Harvey P. Dale, Dec 05 2012 *)

A129798 a(n) = floor(8^n/n).

Original entry on oeis.org

8, 32, 170, 1024, 6553, 43690, 299593, 2097152, 14913080, 107374182, 780903144, 5726623061, 42288908760, 314146179364, 2345624805922, 17592186044416, 132458812569720, 1000799917193443, 7585009898729256

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(8^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

Table[Floor[8^n/n],{n,20}] (* Harvey P. Dale, Aug 03 2019 *)

A129799 a(n) = floor(9^n/n).

Original entry on oeis.org

9, 40, 243, 1640, 11809, 88573, 683281, 5380840, 43046721, 348678440, 2852823600, 23535794706, 195528140640, 1634056603925, 13726075472976, 115813761803240, 981010688215680, 8338590849833284, 71097458824894320

Offset: 1

Mohammad K. Azarian, May 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A000799, A092763, A053638.

[Floor(9^n / n): n in [1..25]]; // Vincenzo Librandi, May 08 2011

A225585 Floor((3^n-1)/n).

Original entry on oeis.org

2, 4, 8, 20, 48, 121, 312, 820, 2186, 5904, 16104, 44286, 122640, 341640, 956593, 2690420, 7596480, 21523360, 61171656, 174339220, 498112057, 1426411800, 4093181688, 11767897353, 33891544377, 97764070320, 282429536480, 817028301962, 2366564736720

Offset: 1

Vincenzo Librandi, May 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A003462, A007051, A024023, A034472, A082482, A092763.

Magma
```
[Floor((3^n-1)/n): n in [1..30]];
```

Table[Floor[(3^n - 1) / n], {n, 30}] (* or *) Table[Quotient[3^n - 1, n], {n, 30}]

A246003 Floor(m^n/n) with n >= m >= 1.

Original entry on oeis.org

1, 0, 2, 0, 2, 9, 0, 4, 20, 64, 0, 6, 48, 204, 625, 0, 10, 121, 682, 2604, 7776, 0, 18, 312, 2340, 11160, 39990, 117649, 0, 32, 820, 8192, 48828, 209952, 720600, 2097152, 0, 56, 2187, 29127, 217013, 1119744, 4483734, 14913080, 43046721, 0, 102, 5904, 104857

Offset: 1

Vincenzo Librandi, Dec 11 2014

			Triangle begins:
1;
0,  2;
0,  2,    9;
0,  4,   20,    64;
0,  6,   48,   204,    625;
0, 10,  121,   682,   2604,    7776;
0, 18,  312,  2340,  11160,   39990,  117649;
0, 32,  820,  8192,  48828,  209952,  720600,  2097152;
0, 56, 2187, 29127, 217013, 1119744, 4483734, 14913080, 43046721;
...

Cf. A117142.

Cf. for the k-th column: A063524 (k=1), A000799(k=2), A092763(k=3), A129794(k=4), A129795(k=5), A129796 (k=6), A129797 (k=7), A129798 (k=8), A129799 (k=9).

/* As triangle: */ [[Floor(m^n/n): m in [1..n]]: n in [1..10]];

Table[Floor[m^n/n], {n, 1, 10}, {m, 1, n}]

tabl(nn) = {for (n=1, nn, for (m=1, n, print1(m^n\n, ", ");); print(););} \\ Michel Marcus, Dec 12 2014

cp's OEIS Frontend

A138256 Smallest positive integer m such that n divides [3^m/m] (=A092763(m)).

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Mathematica

A129795 a(n) = floor(5^n/n).

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Magma

Mathematica

A060155 Table T(n,k) by antidiagonals of floor(n^k/k) [n,k >= 1].

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Formula

A129794 a(n) = floor(4^n/n).

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Python

A129796 a(n) = floor(6^n/n).

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Magma

Mathematica

A129797 a(n) = floor(7^n/n).

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Magma

Mathematica

A129798 a(n) = floor(8^n/n).

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Magma

Mathematica

A129799 a(n) = floor(9^n/n).

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Magma

A225585 Floor((3^n-1)/n).

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