A002379 -id:A002379 - cp's OEIS frontend

A073632 Numbers k such that floor((3/2)^k) = A002379(k) is odd.

0, 1, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 17, 18, 20, 21, 22, 25, 27, 30, 32, 33, 34, 35, 38, 42, 45, 46, 48, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 69, 71, 74, 76, 82, 83, 85, 89, 90, 93, 96, 97, 98, 100, 104, 106, 107, 109, 110, 112, 113, 114, 116, 117, 118, 119, 120

Offset: 1

Benoit Cloitre, Aug 29 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002379, A073633, A073634 (complement).

Select[Range[0, 120], OddQ[Floor[(3/2)^#]] &] (* Jayanta Basu, Jul 03 2013 *)

isok(k) = floor((3/2)^k) % 2; \\ Michel Marcus, May 19 2022

a(1) = 0 inserted by Amiram Eldar, May 19 2022

A073633 Numbers k that divide floor((3/2)^k) = A002379(k).

Original entry on oeis.org

1, 2, 3, 16, 43, 50, 56, 193, 283, 961, 970, 4958, 9439, 10493, 11375, 18552, 57051, 81602, 617287, 917186, 1525995, 5107085, 9162821, 22008620

Offset: 1

Benoit Cloitre, Aug 29 2002

No more terms through 10^6. - Ryan Propper, May 05 2006

The first 8 terms are all in A032863, all known subsequent terms, i.e., at least up to a(21), are not in A032863. - M. F. Hasler, Oct 05 2018

Cf. A002379, A073632, A073634.

t = 1; Do[t = 3t/2; If[ Mod[ Floor[t], n] == 0, Print[n]], {n, 500000}] (* Robert G. Wilson v, Apr 06 2006 *)

a=1;for(n=1,10^6,a*=3;b=shift(a,-n);if(b%n==0,print1(n,","))) \\ Robert Gerbicz, Aug 23 2006

P=1;for(n=1,oo,(P*=3)>>n%n||print1(n",")) \\ M. F. Hasler, Oct 05 2018

from gmpy2 import mpz, t_div_2exp, t_mod
A073633_list, m = [], mpz(1)
for n in range(1,10**9):
    m *= 3
    if t_mod(t_div_2exp(m,n),n) == 0:
        A073633_list.append(n) # Chai Wah Wu, Mar 30 2020

More terms from Michel ten Voorde Jun 20 2003
2 more terms from Ryan Propper, May 05 2006
More terms from Robert Gerbicz, Aug 23 2006
a(22)-a(24) from Chai Wah Wu, Mar 30 2020

A073634 Numbers k such that floor((3/2)^k) = A002379(k) is even.

Original entry on oeis.org

2, 9, 11, 13, 16, 19, 23, 24, 26, 28, 29, 31, 36, 37, 39, 40, 41, 43, 44, 47, 49, 50, 51, 54, 56, 60, 66, 67, 68, 70, 72, 73, 75, 77, 78, 79, 80, 81, 84, 86, 87, 88, 91, 92, 94, 95, 99, 101, 102, 103, 105, 108, 111, 115, 121, 123, 126, 127, 132, 134, 135, 136, 138

Offset: 1

Benoit Cloitre, Aug 29 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002379, A073632 (complement), A073633.

Select[Range[0, 150], EvenQ[Floor[(3/2)^#]] &] (* Amiram Eldar, May 19 2022 *)

Wrong term (0) removed by Amiram Eldar, May 19 2022

A083198 a(n) = A061419(n) - A002379(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 16, 24, 36, 54, 80, 121, 181, 271, 407, 610, 915, 1372, 2058, 3088, 4631, 6947, 10420, 15630, 23446, 35169, 52753, 79129, 118693, 178039, 267059, 400589, 600883, 901324, 1351986, 2027979, 3041968

Offset: 5

Ralf Stephan, Jun 01 2003

lim (a(n) * (2/3)^n) = 2/3 * A083286 - 1.

p=1; for(n=2, 100, p=p+ceil(p/2); print1(p-floor((3/2)^n)", "))

A154130 Exponents m with decreasing fractional part of (4/3)^m.

Original entry on oeis.org

1, 4, 13, 17, 128, 485, 692, 1738, 12863, 77042, 109705, 289047, 720429, 4475944, 75629223, 182575231

Offset: 1

Hieronymus Fischer, Jan 11 2009

nonn

The next term is greater than 3*10^8.

			a(3)=13, since fract((4/3)^13)=0.0923.., but fract((4/3)^k)>=0.16... for 1<=k<=12; thus fract((4/3)^13)<fract((4/3)^k) for 1<=k<13.

Cf. A081464, A153669, A153701, A153708, A154138, A154146.

Cf. A002379, A064628.

Recursion: a(1):=1, a(k):=min{ m>1 | fract((4/3)^m) < fract((4/3)^a(k-1))}, where fract(x) = x-floor(x).

Extended by Charles R Greathouse IV, Nov 03 2009

a(15)-a(16) from Robert Gerbicz, Nov 21 2010

A094969 a(n) = floor(5^n/2^n).

Original entry on oeis.org

1, 2, 6, 15, 39, 97, 244, 610, 1525, 3814, 9536, 23841, 59604, 149011, 372529, 931322, 2328306, 5820766, 14551915, 36379788, 90949470, 227373675, 568434188, 1421085471, 3552713678, 8881784197, 22204460492, 55511151231, 138777878078, 346944695195, 867361737988

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002379, A094970-A094999.

[Floor(5^n / 2^n): n in [0..30]]; // Vincenzo Librandi, Sep 08 2011

A094969:=n->floor(5^n/2^n): seq(A094969(n), n=0..40); # Wesley Ivan Hurt, Feb 22 2017

Table[ Floor[(5/2)^n], {n, 0, 30}]
Floor[(5/2)^Range[0,30]] (* Harvey P. Dale, Jul 16 2012 *)

A094969(n):=floor(5^n/2^n)$ makelist(A094969(n),n,0,30); /* Martin Ettl, Oct 25 2012 */

a(n)=5^n>>n \\ Charles R Greathouse IV, Sep 08 2011

A061419 a(n) = ceiling(a(n-1)*3/2) with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 27, 41, 62, 93, 140, 210, 315, 473, 710, 1065, 1598, 2397, 3596, 5394, 8091, 12137, 18206, 27309, 40964, 61446, 92169, 138254, 207381, 311072, 466608, 699912, 1049868, 1574802, 2362203, 3543305, 5314958, 7972437, 11958656

Offset: 1

Henry Bottomley, May 02 2001

nonn

It appears that this sequence is the (L)-sieve transform of {3,6,9,12,...,3n,...} = A008585. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 06 2009

			a(6) = ceiling(8*3/2) = 12.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 196.

Harry J. Smith, Table of n, a(n) for n = 1..500
Zakir Deniz, Topology of acyclic complexes of tournaments and coloring, Applicable Algebra in Engineering, Communication, March 2015, Volume 26, Issue 1-2, pp. 213-226.
A. Dubickas, On integer sequences generated by linear maps, Glasg. Math. J. 51(2) (2009), 243-252.
Don Knuth, Ambidextrous Numbers, Preprint, September 2022.
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33(2) (1991), 235-240.
Eric Weisstein's World of Mathematics, Power Ceilings.
Index entries for sequences related to the Josephus Problem

Cf. A002379, A003312, A034082, A061418, A061420, A070885, A083286.

First differences are in A073941.

[ n eq 1 select 1 else Ceiling(Self(n-1)*3/2): n in [1..40] ]; // Klaus Brockhaus, Nov 14 2008

a:=proc(n) option remember: if n=1 then 1 else ceil(procname(n-1)*3/2) fi; end; seq(a(n),n=1..40); # Muniru A Asiru, Jun 07 2018

a=1;a=Table[a=Ceiling[a*3/2],{n,0,4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
NestList[Ceiling[3#/2]&,1,39] (* Stefano Spezia, Dec 08 2024 *)

{ a=2/3; for (n=1, 500, write("b061419.txt", n, " ", a=ceil(a*3/2)) ) } \\ Harry J. Smith, Jul 22 2009

from itertools import islice
def A061419_gen(): # generator of terms
    a = 2
    while True:
        yield a-1
        a += a>>1
A061419_list = list(islice(A061419_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n) = A061418(n) - 1 = floor(K*(3/2)^n) where K = 1.08151366859...
The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003
a(1) = 1, a(n) = A070885(n)/3. - Benoit Cloitre, Aug 18 2002
a(n) = ceiling((a(n-1) + a(n-2))*9/10) - Franklin T. Adams-Watters, May 01 2006

A094999 a(n) = floor(12^n/11^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 25, 27, 29, 32, 35, 38, 42, 45, 50, 54, 59, 65, 71, 77, 84, 92, 100, 109, 119, 130, 142, 155, 169, 185, 201, 220, 240, 262, 285, 311, 340, 371, 404, 441, 481

Offset: 0

Robert G. Wilson v, May 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A002379, A094969-A094997.

[Floor(12^n / 11^n): n in [0..80]]; // Vincenzo Librandi, Sep 08 2011

Mathematica
```
Table[ Floor[(12/11)^n], {n, 0, 71}]
```

A094999(n):=floor(12^n/11^n)$ makelist(A094999(n),n,0,60); /* Martin Ettl, Oct 25 2012 */

a(n)=12^n\11^n \\ Charles R Greathouse IV, Feb 04 2016

It appears that a(n) = floor(2^(n/8)). - Yakov Shusterman (yakov(AT)mobli.com), Dec 07 2008

A061418 a(n) = floor(a(n-1)*3/2) with a(1) = 2.

Original entry on oeis.org

2, 3, 4, 6, 9, 13, 19, 28, 42, 63, 94, 141, 211, 316, 474, 711, 1066, 1599, 2398, 3597, 5395, 8092, 12138, 18207, 27310, 40965, 61447, 92170, 138255, 207382, 311073, 466609, 699913, 1049869, 1574803, 2362204, 3543306, 5314959, 7972438

Offset: 1

Henry Bottomley, May 02 2001

Can be stated as the number of animals starting from a single pair if any pair of animals can produce a single offspring (as in the game Minecraft, if the player allows offspring to fully grow before breeding again). - Denis Moskowitz, Dec 05 2012

Maximum number of partial products that can be added in a Wallace tree multiplier with n-1 full adder stages. - Chinmaya Dash, Aug 19 2020

			a(6) = floor(9*3/2) = 13.

Iain Fox, Table of n, a(n) for n = 1..1000 (first 500 terms from Harry J. Smith)
Don Knuth, Ambidextrous Numbers, Preprint, September 2022.
M. van de Vel, Determination of msd(L^n), J. Algebraic Combin. 9(2) (1999), 161-171. See Table 5. - _N. J. A. Sloane_, Mar 26 2012
Mark van Wijk, The Quest for the Best Thread-Safe Java List, Univ. of Twente (Netherlands 2022).
Wikipedia, Wallace tree.

Cf. A002379, A003312, A034082, A061419, A083286.

First differences are in A073941.

[ n eq 1 select 2 else Floor(Self(n-1)*(3/2)): n in [1..39] ]; // Klaus Brockhaus, Nov 14 2008

{ a=4/3; for (n=1, 500, a=a*3\2; write("b061418.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 22 2009

first(n) = my(v=vector(n)); v[1]=2; for(i=2, n, v[i]=v[i-1]*3\2); v \\ Iain Fox, Jul 15 2022

from itertools import islice
def A061418_gen(): # generator of terms
    a = 2
    while True:
        yield a
        a += a>>1
A061418_list = list(islice(A061418_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n) = A061419(n) + 1 = ceiling(K*(3/2)^n) where K = 1.08151366859...

The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003

A002380 a(n) = 3^n reduced modulo 2^n.

Original entry on oeis.org

0, 1, 1, 3, 1, 19, 25, 11, 161, 227, 681, 1019, 3057, 5075, 15225, 29291, 55105, 34243, 233801, 439259, 269201, 1856179, 3471385, 6219851, 1882337, 5647011, 50495465, 17268667, 186023729, 21200275, 63600825, 1264544299, 3793632897, 7085931395

Offset: 0

N. J. A. Sloane

A065554 lists the indices n such that a(n+1) = 3*a(n). - Benoit Cloitre, Apr 21 2003

a(n) = (fractional part of (3/2)^n without the decimal point)/5^n = A204544(n) / 5^n. - Michel Lagneau, Jan 25 2012

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3322 (first 101 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Fractional Part.
Eric Weisstein's World of Mathematics, Power Fractional Parts.

Cf. A060692, A002379, A000079, A000244.

Cf. k^n mod (k-1)^n: this sequence (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

a002380 n = 3^n `mod` 2^n  -- Reinhard Zumkeller, Jul 11 2014

a:=n->3^n mod(2^n): seq(a(n), n=0..33); # Zerinvary Lajos, Feb 15 2008

Table[ PowerMod[3, n, 2^n], {n, 0, 33}] (* Robert G. Wilson v, Dec 14 2006 *)
Table[ 3^n - 2^n * Floor[ (3/2)^n ], {n,0,33} ] (* Fred Daniel Kline, Oct 12 2017 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi;
y[n_] := 3^n - 2^n * x[n];
Array[y, 33] (* Fred Daniel Kline, Dec 21 2017 *)

concat([0],vector(55,n,lift(Mod(3,2^n)^n))) \\ Joerg Arndt, Oct 14 2017

More terms from Jason Earls, Jul 29 2001

cp's OEIS Frontend

A073632 Numbers k such that floor((3/2)^k) = A002379(k) is odd.

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A073633 Numbers k that divide floor((3/2)^k) = A002379(k).

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A073634 Numbers k such that floor((3/2)^k) = A002379(k) is even.

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A083198 a(n) = A061419(n) - A002379(n).

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A154130 Exponents m with decreasing fractional part of (4/3)^m.

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A094969 a(n) = floor(5^n/2^n).

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A061419 a(n) = ceiling(a(n-1)*3/2) with a(1) = 1.

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A094999 a(n) = floor(12^n/11^n).

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