A034082 -id:A034082 - cp's OEIS frontend

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

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

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

A034062 Decimal part of a(n)^(1/7) starts with n (7th powers excluded).

Original entry on oeis.org

129, 2, 4, 7, 11, 18, 27, 42, 62, 90, 2, 187, 193, 199, 206, 213, 3, 227, 234, 242, 250, 4, 266, 275, 283, 5, 302, 311, 321, 6, 341, 351, 7, 373, 8, 396, 9, 420, 10, 446, 11, 473, 12, 501, 13, 14, 546, 15, 16, 17, 611, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29

Offset: 0

Patrick De Geest, Sep 15 1998

			a(0)=129 -> 129^(1/7)=2.{0}022247...: a(1)=2 -> 2^(1/7)=1.{1}040895...

Cf. A034072, A034082.

A034072 Decimal part of a(n)^(1/7) starts with n so that a(n) < a(n+1).

Original entry on oeis.org

129, 181, 250, 341, 459, 611, 804, 1047, 1350, 1725, 2752, 2814, 2878, 2944, 3010, 3078, 3147, 3217, 3289, 3362, 3436, 3512, 3590, 3668, 3749, 3830, 3914, 3998, 4085, 4173, 4262, 4354, 4446, 4541, 4637, 4735, 4835, 4937, 5040, 5146, 5253, 5362, 5473

Offset: 0

Patrick De Geest, Sep 15 1998

			a(11)=2814 -> 2814^(1/7)=3.{11}000140...;
a(12)=2878 -> 2878^(1/7)=3.{12}000885... and a(11)=2814 < 2878.

Cf. A034062, A034082.

A334264 Numbers k > 1 such that (3/2)^k sets a new record for closest fractional part to 1/2.

Original entry on oeis.org

2, 3, 5, 9, 11, 69, 420, 2361, 12432, 21565, 28226, 128389, 274555, 497269, 836000, 1151341, 1973112, 2202332, 2458844, 5402520

Offset: 1

Ben Paul Thurston, Apr 20 2020

A081464 is also related to (3/2) to a power being a record distance from a value of an integer.

Cf. A002379, A034082, A179523.

off := 1; for i from 2 to 1000 do t := (1+1/2)^i-floor((1+1/2)^i); d := abs(1/2-t); if d < off then off := d; print(i) end if end do

dm = 1; r = 3/2; s = {}; Do[r *= 3/2; If[(d = Abs[r - Floor[r] - 1/2]) < dm, dm = d; AppendTo[s, n + 1]], {n, 1, 10^7}]; s (* Amiram Eldar, Jun 08 2020 *)

a(8)-a(13) from Amiram Eldar, Jun 08 2020
a(14)-a(16) from Chai Wah Wu, Jul 02 2020
a(17)-a(20) from Bert Dobbelaere, Apr 25 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

PARI

Python

Formula

A034062 Decimal part of a(n)^(1/7) starts with n (7th powers excluded).

Views

Author

Keywords

Examples

Crossrefs

A034072 Decimal part of a(n)^(1/7) starts with n so that a(n) < a(n+1).

Views

Author

Keywords

Examples

Crossrefs

A334264 Numbers k > 1 such that (3/2)^k sets a new record for closest fractional part to 1/2.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Extensions