A000799 -id:A000799 - cp's OEIS frontend

A138255 Smallest positive integer m such that n divides [2^m/m] (=A000799(m)).

1, 1, 5, 4, 6, 5, 9, 8, 7, 6, 12, 15, 14, 9, 13, 8, 10, 7, 21, 28, 13, 24, 48, 15, 22, 14, 19, 9, 30, 13, 11, 8, 31, 10, 13, 21, 38, 21, 14, 39, 22, 13, 29, 63, 13, 67, 135, 65, 43, 22, 10, 15, 35, 19, 24, 9, 21, 30, 120, 28, 62, 11, 13, 16, 14, 31, 69, 20, 67, 13, 145, 21, 19, 38

Offset: 1

Max Alekseyev, Mar 09 2008

nonn

This sequence is well-defined.

Robert Israel, Table of n, a(n) for n = 1..10000
Romanian Master in Mathematics Contest, Problem 3, Bucharest, 2007.

Cf. A000799, A138256, A138257, A138258, A138259, A138260, A138261, A138262, A138263.

f:= proc(n) local m;
     for m from 1 do if floor(2^m/m) mod n = 0 then return m fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 07 2018

a[n_] := For[m = 1, True, m++, If[Divisible[Floor[2^m/m], n], Return[m]]];
Array[a, 100] (* Jean-François Alcover, Mar 22 2019 *)

from itertools import count
def A138255(n): return next(m for m in count(1) if not (1<Chai Wah Wu, Aug 24 2023

A114699 A000799(n) - A064355(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 6, 0, 0, 10, 1, 0, 18, 0, 0, 36, 0, 0, 62, 0, 4, 113, 0, 0, 210, 6, 0, 391, 0, 0, 739, 0, 0, 1365, 2, 20, 2570, 0, 0, 4854, 37, 4, 9198, 0, 0, 17544, 0, 0, 33296, 0, 126, 63550, 0, 0, 121574, 248, 0, 233016, 0, 0, 447828, 0

Offset: 0

Paul D. Hanna, Feb 20 2006

nonn

There was a suggestion that these were the same sequence, but they are not.

Cf. A000799, A064355.

f[n_] := Block[{d = Select[Divisors@n, OddQ@# &]}, Floor[2^n/n] - Plus @@ (2^(n/d)*MoebiusMu@d)/n]; Array[f, 76] (* Robert G. Wilson v *)

A000799(n) = A064355(n) just for: (1) n a power of 2. (2) n = p * 2^k, where p is a prime with p > 2^{2^k - k}. (Including the case k=0. ) (3) n = 9. -Franklin T. Adams-Watters, Feb 14 2006

More terms from Robert G. Wilson v, Feb 20 2006

A092763 a(n) = floor(3^n / n).

Original entry on oeis.org

3, 4, 9, 20, 48, 121, 312, 820, 2187, 5904, 16104, 44286, 122640, 341640, 956593, 2690420, 7596480, 21523360, 61171656, 174339220, 498112057, 1426411800, 4093181688, 11767897353, 33891544377, 97764070320, 282429536481

Offset: 1

Reinhard Zumkeller, Apr 13 2004

nonn

a(n) = (A000244(n) - A066601(n))/n.

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

Cf. A000799.

[Floor(3^n / n): n in [1..45]]; // Vincenzo Librandi, May 08 2011

Table[Floor[3^n/n],{n,30}] (* Harvey P. Dale, Jun 13 2013 *)

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 *)

A003176 Integer part of 24(2^n-1)/n.

Original entry on oeis.org

24, 36, 56, 90, 148, 252, 435, 765, 1362, 2455, 4466, 8190, 15121, 28085, 52427, 98302, 185041, 349524, 662257, 1258290, 2396744, 4575603, 8753329, 16777215, 32212253, 61946642, 119304646, 230087532

Offset: 1

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300
R. J. Penrose, Puzzle, Twistor Newsletter, No. 10 (July 1980), p. 22.
R. J. Penrose, Puzzle, Twistor Newsletter, No. 10 (July 1980), p. 22. [Cached copy]
R. J. Penrose, Solution to puzzle, Twistor Newsletter, No. 41, p. 37, 1996.
R. J. Penrose, Solution to puzzle, Twistor Newsletter, No. 41, p. 37, 1996. [Cached copy]

Cf. A000799, A003138, A003177, A121056.

[Floor(24*(2^n-1)/n): n in [1..45]]; // Vincenzo Librandi, May 08 2011

a[n_] := Floor[24*(2^n-1)/n]; Array[a, 30] (* Giovanni Resta, Mar 25 2017 *)

def A003176(n): return ((3<Chai Wah Wu, Aug 24 2023

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.

A060156 a(n) = floor(10^n/n).

Original entry on oeis.org

10, 50, 333, 2500, 20000, 166666, 1428571, 12500000, 111111111, 1000000000, 9090909090, 83333333333, 769230769230, 7142857142857, 66666666666666, 625000000000000

Offset: 1

Henry Bottomley, Mar 12 2001

nonn

			a(6) = floor(1000000/6) = 166666.

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A000799, A056159, A060155.

{ default(realprecision, 10); for (n=1, 200, write("b060156.txt", n, " ", floor(10^n/n)); ) } \\ Harry J. Smith, Jul 02 2009

a(n) = (A011557(n) - A056969(n))/n.

A082493 a(n) = n*ceiling(2^n/n) - 2^n.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 5, 0, 1, 6, 9, 8, 11, 10, 7, 0, 15, 8, 17, 4, 13, 18, 21, 8, 18, 22, 1, 12, 27, 26, 29, 0, 25, 30, 17, 8, 35, 34, 31, 24, 39, 20, 41, 28, 28, 42, 45, 32, 19, 26, 43, 36, 51, 26, 12, 24, 49, 54, 57, 44, 59, 58, 55, 0, 33, 2, 65, 52, 61, 26, 69, 8, 71, 70, 7, 60, 59, 14

Offset: 1

Vladeta Jovovic, Apr 28 2003

Least nonnegative k such that (2^n+k)/n is an integer.

If n is a power of 2, a(n) = 0; otherwise a(n) = n - A015910(n). - Robert Israel, Apr 08 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053638, A015910, A000799, A082494, A215747.

seq(-2&^n mod n, n = 1 .. 100); # Robert Israel, Apr 08 2015

def A082493(n): return (-pow(2,n,n))%n # Chai Wah Wu, Aug 24 2023

a(n) = -(2^n) mod n. - Robert Israel, Apr 08 2015

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 *)

cp's OEIS Frontend

A138255 Smallest positive integer m such that n divides [2^m/m] (=A000799(m)).

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A114699 A000799(n) - A064355(n).

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A092763 a(n) = floor(3^n / n).

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Magma

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

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Magma

Mathematica

A003176 Integer part of 24(2^n-1)/n.

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Magma

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

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A060156 a(n) = floor(10^n/n).

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PARI

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A082493 a(n) = n*ceiling(2^n/n) - 2^n.

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