A060692 -id:A060692 - cp's OEIS frontend

A064464 Binary order (cf. A029837) of the number of parts if 3^n is partitioned into parts of size 2^n as far as possible and into parts of size 1^n (cf. A060692).

1, 2, 3, 3, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 18, 19, 19, 21, 22, 23, 21, 23, 26, 25, 28, 25, 26, 31, 32, 33, 34, 35, 35, 37, 38, 39, 39, 40, 42, 43, 44, 44, 46, 47, 47, 47, 48, 50, 51, 51, 54, 54, 56, 56, 58, 59, 60, 60, 59, 63, 63, 63, 66, 65, 67, 69, 69, 70, 69

Offset: 1

Labos Elemer, Oct 03 2001; revised Mar 10 2002

nonn

These binary orders are nearly equal to n.

For several values of n, a(n) = n holds, e.g., for n = 1, 2, 3, 5, 6, 8, 9, 10, 11,12.

			For n=12, 3^12 = 531441 = 129*2^12 + 3057*1^12; the binary order of 129 + 3057 = 3186 is ceiling(log_2(3186)) = 12, the exponent.

Cf. A029837, A060692, A064630.

{for(n=1,72,d=divrem(3^n,2^n); print1(ceil(log(d[1]+d[2])/log(2)),","))}

a(n) = A029837(A060692(n)) = ceiling(log_2(A060692(n))).

Edited by Klaus Brockhaus, May 24 2003

A002379 a(n) = floor(3^n / 2^n).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 25, 38, 57, 86, 129, 194, 291, 437, 656, 985, 1477, 2216, 3325, 4987, 7481, 11222, 16834, 25251, 37876, 56815, 85222, 127834, 191751, 287626, 431439, 647159, 970739, 1456109, 2184164, 3276246, 4914369, 7371554, 11057332

Offset: 0

N. J. A. Sloane

It is an important unsolved problem related to Waring's problem to show that a(n) = floor((3^n-1)/(2^n-1)) holds for all n > 1. This has been checked for 10000 terms and is true for all sufficiently large n, by a theorem of Mahler. [Lichiardopol]
a(n) = floor((3^n-1)/(2^n-1)) holds true at least for 2 <= n <= 305000. - Hieronymus Fischer, Dec 31 2008
a(n) is also the curve length (rounded down) of the Sierpiński arrowhead curve after n iterations, let a(0) = 1. - Kival Ngaokrajang, May 21 2014
a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 5, 7 or 11 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

R. K. Guy, Unsolved Problems in Number Theory, E19.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. Lichiardopol, Problem 925 (BCC20.19), A number-theoretic problem, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
K. Mahler, On the fractional parts of the powers of a rational number, II, Mathematika 4 (1957), 122-124.
Kival Ngaokrajang, Illustration of Sierpinski arrowhead curve for n = 0..5
Eric Weisstein's World of Mathematics, Power Floors
Wikipedia, Sierpiński arrowhead curve

Cf. A094969-A094500, A000217, A081464, A153662, A153665, A153666.
Cf. A060692, A002380, A000079, A000244.
Cf. A046037, A070758, A070759, A067904 (Composites and Primes).
Cf. A064628 (an analog for 4/3).

a002379 n = 3^n `div` 2^n  -- Reinhard Zumkeller, Jul 11 2014

[Floor(3^n / 2^n): n in [0..40]]; // Vincenzo Librandi, Sep 08 2011

A002379:=n->floor(3^n/2^n); seq(A002379(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[(3/2)^n], {n, 0, 40}] (* Robert G. Wilson v, May 11 2004 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; Array[x, 40] (* Fred Daniel Kline, Dec 21 2017 *)
x[n_]:=Round[-(1/2) + (3^n - 1)/(2^n - 1)]; Array[x, 39, 2] (* offset n+1, Fred Daniel Kline, Apr 13 2018 *)

makelist(floor(3^n/2^n), n, 0, 50); /* Martin Ettl, Oct 17 2012 */

a(n)=3^n>>n \\ Charles R Greathouse IV, Jun 10 2011

def A002379(n): return 3**n>>n # Chai Wah Wu, Sep 21 2022

a(n) = b(n) - (-2/3)^n where b(n) is defined by the recursion b(0):=2, b(1):=5/6, b(n+1):=(5/6)*b(n) + b(n-1). - Hieronymus Fischer, Dec 31 2008
a(n) = (1/2)*(b(n) + sqrt(b(n)^2 - (-4)^n)) (with b(n) as defined above). - Hieronymus Fischer, Dec 31 2008
3^n = a(n)*2^n + A002380(n). - R. J. Mathar, Oct 26 2012
a(n) = -(1/2) + (3/2)^n + arctan(cot((3/2)^n Pi)) / Pi. - Fred Daniel Kline, Apr 14 2018
a(n+1) = round( -(1/2) + (3^n-1)/(2^n-1) ). - Fred Daniel Kline, Apr 14 2018

More terms from Robert G. Wilson v, May 11 2004

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

A064628 a(n) = floor((4/3)^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, 31, 42, 56, 74, 99, 133, 177, 236, 315, 420, 560, 747, 996, 1328, 1771, 2362, 3149, 4199, 5599, 7466, 9954, 13273, 17697, 23596, 31462, 41950, 55933, 74577, 99437, 132583, 176777, 235703, 314271, 419028, 558704

Offset: 0

Labos Elemer, Oct 01 2001

a(n) is the perimeter of a hexaflake (rounded down) after n iterations. The total number of holes = A000420(n) - 1. The total number of irregular polygon holes = A000420(n-1) - 1. The total number of triangle holes = 6*A000420(n-1). - Kival Ngaokrajang, Apr 18 2014

a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 3, 5 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

R. K. Guy, Unsolved Problems in Number Theory, E19.

Harry J. Smith, Table of n, a(n) for n=0,...,400
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
Kival Ngaokrajang, Illustration of hexaflake for n = 0..3.
Eric Weisstein's World of Mathematics, Power Floors.
Wikipedia, n-flake.

Cf. A002379, A002380, A060692.
Cf. A094969 - A094500.
Cf. A046038, A070761, A070762, A067905 (Composites and Primes).

A064628:=n->floor(4^n/3^n); seq(A064628(n), n=0..30); # Wesley Ivan Hurt, Apr 19 2014

Table[Floor[(4/3)^n], {n, 0, 30}] (* Robert G. Wilson v *)

{ f=t=1; for (n=0, 400, write("b064628.txt", n, " ", f\t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

def A064628(n): return floor((4/3)^n)
[A064628(n) for n in range(0, 46)] # Stefano Spezia, Oct 13 2024

More terms from Robert G. Wilson v, May 26 2004

OFFSET changed from 1 to 0 by Harry J. Smith, Sep 20 2009

A064629 a(n) = 4^n mod 3^n.

Original entry on oeis.org

0, 1, 7, 10, 13, 52, 451, 1075, 6487, 6265, 44743, 119923, 302545, 147298, 589192, 11922706, 33341917, 4227505, 146050183, 584200732, 1174541461, 4698165844, 18792663376, 43789593895, 175158375580, 700633502320, 1955245399837, 2737249942690, 18574597255747

Offset: 0

Labos Elemer, Oct 01 2001

(a(n+1) - 4*a(n))/3^n is always one of -3, -2, -1, 0, 1, 2. - Robert Israel, Dec 01 2016

Harry J. Smith and Seiichi Manyama, Table of n, a(n) for n = 0..2096 (first 201 terms from Harry J. Smith)

Cf. A002379, A060692, A064628.

Cf. k^n mod (k-1)^n: A002380 (k=3), this sequence (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

[4^n mod 3^n: n in [0..40]]; // Wesley Ivan Hurt, Dec 01 2016

A064629:=n->4^n mod 3^n: seq(A064629(n), n=0..40); # Wesley Ivan Hurt, Dec 01 2016

Table[PowerMod[4,n,3^n],{n,0,30}] (* Harvey P. Dale, Aug 26 2012 *)

{ f=t=1; for (n=0, 200, write("b064629.txt", n, " ", f%t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

[power_mod(4,n,3^n)for n in range(0,27)] # Zerinvary Lajos, Nov 28 2009

a(26) from Harry J. Smith, Sep 20 2009

A064630 Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.

Original entry on oeis.org

2, 5, 5, 16, 25, 15, 66, 121, 146, 771, 1220, 3641, 8093, 15843, 28359, 50236, 33366, 36709, 145250, 137776, 548024, 2186496, 1066102, 4251976, 16984368, 28678103, 13620614, 205950171, 100716646, 381399635, 1397934923, 3826001641

Offset: 1

Labos Elemer, Oct 01 2001

nonn

Corresponds to the only solution of the Diophantine equation 4^n = x*3^n + y*2^n + z*1^n with constraints 0 <= y < 3^n/2^n, 0 <= z < 2^n.

Binary order (cf. A029837) of a(n) is close to n.

			4^6 = 4096 = 729 + 729 + 729 + 729 + 729 + 64 + 64 + 64 + 64 + 64 + 64 + 64 + 1 + 1 + 1 = 5*3^6 + 7*2^6 + 3*1^6, so a(6) = 5 + 7 + 3 = 15.

Harry J. Smith, Table of n, a(n) for n = 1..250

Cf. A064628, A064629, A060692, A029837.

{for(n=1,32,a=divrem(4^n,3^n); b=divrem(a[2],2^n); print1(a[1]+b[1]+b[2],","))}

{ f=t=w=1; for (n=1, 250, f*=4; t*=3; w*=2; a=divrem(f, t); b=divrem(a[2], w); write("b064630.txt", n, " ", a[1]+b[1]+b[2]) ) } \\ Harry J. Smith, Sep 20 2009

a(n) = A064628(n) + floor(A064629(n)/2^n) + (A064629(n) mod 2^n) = floor(4^n/3^n) + floor((4^n mod 3^n)/2^n) + ((4^n mod 3^n) mod 2^n)

Edited by Klaus Brockhaus, May 24 2003

A064536 a(n) = (4^n mod 3^n) mod 2^n.

Original entry on oeis.org

1, 3, 2, 13, 20, 3, 51, 87, 121, 711, 1139, 3537, 8034, 15752, 27922, 49629, 33201, 35975, 143900, 136341, 545364, 2181456, 1060135, 4240540, 16962160, 28647197, 13597858, 205877827, 100616667, 381266393, 1397863922, 3825576990, 8216376565, 14181633879, 22366797148

Offset: 1

Labos Elemer, Oct 08 2001

nonn

A generalization of A002380. It arises also as a coefficient (=c1) of 1^n=1 in a special (greedy) decomposition of 4^n into like powers as follows: 4^n = c3*3^n + c2*2^n + c1*1^n.

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

Cf. A002380, A064853-A064855, A060692, A064628-A064631.

Table[Mod[PowerMod[4,n,3^n],2^n],{n,40}] (* Harvey P. Dale, Apr 09 2013 *)

a(n) = { (4^n % 3^n) % 2^n } \\ Harry J. Smith, Sep 17 2009

n = 7: 4^7 = 16384 = 7*2187 + 8*128 + 51*1 where a(7)=51, the last coefficient; A064630(7) = 7 + 8 + a(7) = 66.

A064631 a(n) = ceiling(log_2(A064630(n))).

Original entry on oeis.org

2, 3, 3, 5, 5, 4, 7, 7, 8, 10, 11, 12, 13, 14, 15, 16, 16, 16, 18, 18, 20, 22, 21, 23, 25, 25, 24, 28, 27, 29, 31, 32, 33, 34, 35, 36, 37, 37, 39, 40, 39, 42, 42, 44, 44, 46, 46, 46, 49, 50, 51, 51, 51, 54, 55, 55, 57, 57, 59, 60, 60, 61, 63, 64, 64, 66, 60, 62, 67, 70, 69, 72

Offset: 1

Labos Elemer, Oct 01 2001

nonn

In A064630, using a greedy algorithm we write 4^n = x*3^n+y*2^n+z*1^n and A064630(n) = x+y+z. This sequence is a measure of the "length" or complexity of those solutions.

Cf. A002379, A002380, A060692, A064628, A064629, A064630, A029837.

a(n) = A029837(A064630(n)) = ceiling(log_2(A064630(n))).

Initial terms corrected and entry revised by Sean A. Irvine, Jul 18 2023

A064855 a(n) = (((6^n mod 5^n) mod 4^n) mod 3^n) mod 2^n.

Original entry on oeis.org

1, 2, 0, 14, 16, 10, 66, 21, 321, 917, 2037, 1550, 2420, 15152, 27439, 46731, 110953, 137148, 336949, 703202, 805647, 181132, 5835407, 3343039, 21816283, 18528238, 95129681, 241918238, 311938330, 48698222, 1539688558, 3481498150, 8104918325, 13512884439, 22365723609

Offset: 1

Labos Elemer, Oct 08 2001

nonn

A generalization of A002380, A064536 and A064854. It arises also as a coefficient (=c1) of 1^n=1 in a special (greedy) decomposition of 6^n into like powers as follows: 6^n = c5*5^n + c4*4^n + c3*3^n + c2*2^n + c1*1^n.

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

Cf. A002380, A064536, A064854, A064855, A060692, A064628-A064631.

Table[Fold[Mod,6^n,Range[5,2,-1]^n],{n,40}]  (* Harvey P. Dale, Mar 14 2011 *)

a(n) = { (((6^n%5^n)%4^n)%3^n)%2^n } \\ Harry J. Smith, Sep 28 2009

n = 8: 6^8 = 1679616 = 4*390625 + 1*65536 + 7*6561 + 22*256 + 21*1 where a(8)=21, the last coefficient and here 6^8 is decomposed into 4 + 1 + 7 + 22 + 21 = 55 like (8th) powers.

A064854 a(n) = ((5^n mod 4^n) mod 3^n) mod 2^n.

Original entry on oeis.org

1, 0, 7, 0, 21, 37, 118, 56, 19, 428, 808, 3920, 2256, 15240, 28312, 46733, 128931, 251439, 434788, 645833, 1397733, 1179155, 7185704, 1551886, 33308648, 65879944, 121274199, 65829274, 228529703, 248939750, 799831532, 2835988891, 1358930753, 9419331043, 9093076436

Offset: 1

Labos Elemer, Oct 08 2001

nonn

A generalization of A002380 and A064536. It arises also as a coefficient (=c1) of 1^n=1 in a special (greedy) decomposition of 5^n into like powers as follows: 5^n = c4*4^n + c3*3^n + c2*2^n + c1*1^n.

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

Cf. A002380, A064536, A064855, A060692, A064628-A064631.

a(n) = { ((5^n%4^n)%3^n)%2^n } \\ Harry J. Smith, Sep 28 2009

n = 7: 5^7 = 78125 = 4*16384 + 5*2187 + 12*128 + 118*1, where a(7)=118, the last coefficient.

cp's OEIS Frontend

A064464 Binary order (cf. A029837) of the number of parts if 3^n is partitioned into parts of size 2^n as far as possible and into parts of size 1^n (cf. A060692).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A002379 a(n) = floor(3^n / 2^n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

A064628 a(n) = floor((4/3)^n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Extensions

A064629 a(n) = 4^n mod 3^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A064630 Number of parts if 4^n is partitioned into parts of size 3^n as far as possible into parts of size 2^n as far as possible and into parts of size 1^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs