A002380 -id:A002380 - cp's OEIS frontend

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

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

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

A060692 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.

Original entry on oeis.org

2, 3, 6, 6, 26, 36, 28, 186, 265, 738, 1105, 3186, 5269, 15516, 29728, 55761, 35228, 235278, 441475, 272526, 1861166, 3478866, 6231073, 1899171, 5672262, 50533341, 17325482, 186108951, 21328109, 63792576, 1264831925, 3794064336, 7086578554

Offset: 1

Labos Elemer, Apr 20 2001

Corresponds to the only solution of the Diophantine equation 3^n = x*2^n + y*1^n with constraint 0 <= y < 2^n. (Since 3^n is odd, of course y cannot be zero.)

			3^4 = 81 = 16 + 16 + 16 + 16 + 16 + 1, so a(4) = 5 + 1 = 6;
3^5 = 243 = 32 + 32 + 32 + 32 + 32 + 32 + 32 + 19*1, so a(5) = 7 + 19 = 26.

Iain Fox, Table of n, a(n) for n = 1..3322 (first 500 terms from Harry J. Smith)

Cf. A002379, A002380, A064464, A064630.

Cf. A000079, A000244.

a060692 n = uncurry (+) $ divMod (3 ^ n) (2 ^ n)
-- Reinhard Zumkeller, Jul 11 2014

Table[3^n - (-1 + 2^n) Floor[(3/2)^n], {n, 33}] (* Fred Daniel Kline, Nov 01 2017 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi;
y[n_] := 3^n - 2^n * x[n]; yplusx[n_] := y[n] + x[n];
Array[yplusx, 33] (* Fred Daniel Kline, Dec 21 2017 *)
f[n_] := Floor[3^n/2^n] + PowerMod[3, n, 2^n]; Array[f, 33] (* Robert G. Wilson v, Dec 27 2017 *)

a(n) = { my(d=divrem(3^n,2^n)); d[1]+d[2] }

a(n) = { (3^n\2^n) + (3^n%2^n) } \\ Harry J. Smith, Jul 09 2009

a(n) = A002379(n) + A002380(n) = floor(3^n/2^n) + (3^n mod 2^n).

For n > 2, a(n) = 3^n mod (2^n-1). - Alex Ratushnyak, Jul 22 2012

Edited by Klaus Brockhaus, May 24 2003

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

A065554 Numbers k such that floor((3/2)^(k+1))/floor((3/2)^k) = 3/2.

Original entry on oeis.org

2, 9, 11, 13, 24, 29, 31, 36, 37, 40, 41, 43, 49, 50, 51, 67, 68, 70, 72, 73, 77, 79, 80, 86, 88, 91, 92, 95, 101, 102, 103, 115, 121, 126, 127, 132, 134, 136, 142, 145, 146, 151, 154, 156, 162, 165, 167, 171, 172, 176, 178, 179, 181, 191, 193, 194, 195, 198, 199

Offset: 1

Benoit Cloitre, Nov 28 2001

nonn

Also k such that A002380(k+1) = 3*A002380(k). - Benoit Cloitre, Apr 21 2003

It appears that lim_{n->oo} a(n)/n = 3. - Benoit Cloitre, Jan 29 2006

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002379, A002380.

a[1] = 2; a[n_ ] := a[n] = Block[ {k = a[n - 1] + 1}, While[ Floor[(3/2)^(k + 1)] / Floor[(3/2)^k] != 3/2, k++ ]; Return[k]]; Table[ a[n], {n, 1, 70} ]

isok(k) = { my(f=3/2); floor(f^(k+1))/floor(f^k) == f } \\ Harry J. Smith, Oct 22 2009

More terms from Robert G. Wilson v, Nov 30 2001

A138589 a(n) = 5^n mod 4^n.

Original entry on oeis.org

0, 1, 9, 61, 113, 53, 3337, 12589, 62945, 118117, 328441, 2690781, 9259601, 12743573, 197935593, 452807053, 2264035265, 7025209029, 35126045145, 106910748989, 809431651889, 1848135003893, 9240675019465, 28611189052909, 213424689442209, 785648470500389

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1661

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), this sequence (k=5), A138649 (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

Table[PowerMod[5, n, 4^n], {n,0,50}] (* G. C. Greubel, Oct 01 2017 *)

concat([0], vector(50, n, lift(Mod(5, 4^n)^n))) \\ Michel Marcus, Oct 02 2017

A138649 a(n) = 6^n mod 5^n.

Original entry on oeis.org

0, 1, 11, 91, 46, 1526, 15406, 45561, 117116, 312071, 1872426, 21000181, 223657336, 853662766, 5121976596, 12421312701, 74527876206, 141991475986, 2377827762166, 18081663838621, 32196037719226, 2441363034106, 491485336407761, 2948912018446566

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1430

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), this sequence (k=6), A139786 (k=7), A138973 (k=8), A139733 (k=9).

Table[PowerMod[6,n,5^n],{n,0,30}] (* Harvey P. Dale, Jan 23 2014 *)

A138973 a(n) = 8^n mod 7^n.

Original entry on oeis.org

0, 1, 15, 169, 1695, 15961, 26846, 450066, 5247614, 13156907, 226316077, 680627620, 13354327932, 65310761853, 328708074010, 1951441519231, 15611532153848, 158125187800385, 101848932467045, 7328445851378156, 35829776440278962, 286638211522231696

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1183

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), this sequence (k=8), A139733 (k=9).

Cf. A000420, A001018.

a[n_]:=PowerMod[8,n,7^n];Array[a,22,0] (* James C. McMahon, Jun 23 2025 *)

a(n) = lift(Mod(8, 7^n)^n); \\ Michel Marcus, Feb 20 2018

[power_mod(8,n,7^n) for n in range(0,22)] # Zerinvary Lajos, Nov 28 2009

A139786 a(n) = 7^n mod 6^n.

Original entry on oeis.org

0, 1, 13, 127, 1105, 1255, 24337, 263671, 725953, 42823, 40610545, 163341463, 780593185, 5464152295, 51309760081, 45711664183, 2200721587585, 12583941205639, 3454291215793, 430419865184215, 3012939056289505, 10122098073837607, 92791637157241105, 517919756258420599

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1285

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), this sequence (k=7), A138973 (k=8), A139733 (k=9).

A264918 Decimal expansion of constant z = Sum_{n>=1} {(3/2)^n} / 2^n, where {x} denotes the fractional part of x.

Original entry on oeis.org

3, 9, 3, 1, 8, 8, 4, 7, 7, 0, 4, 9, 6, 4, 4, 3, 2, 4, 4, 9, 7, 2, 5, 8, 2, 1, 3, 1, 3, 8, 9, 0, 3, 8, 8, 5, 8, 5, 4, 8, 3, 9, 1, 4, 0, 7, 8, 8, 6, 6, 2, 8, 6, 9, 5, 3, 9, 2, 9, 3, 2, 4, 7, 5, 7, 5, 7, 8, 7, 7, 5, 8, 3, 3, 8, 9, 7, 4, 9, 8, 6, 6, 8, 1, 0, 9, 7, 6, 6, 6, 2, 0, 6, 1, 0, 1, 8, 5, 8, 8, 8, 0, 1, 3, 3, 3, 0, 0, 8, 0, 5, 9, 3, 2, 2, 6, 3, 1, 5, 3, 2, 6, 8, 0, 9, 0, 4, 7, 5, 0, 4, 9, 4, 2, 6, 6, 6, 1, 2, 1, 1, 4, 2, 4, 3, 3, 4, 9, 8, 4, 4, 3, 5, 8, 4, 7, 7, 5, 8, 5, 0, 6, 5, 5, 9, 3, 3, 7, 2, 5, 0, 9, 1, 4, 3, 2, 8, 8, 7, 7, 0, 5, 4, 3, 2, 2, 3, 1, 4, 0, 7, 7, 1, 7, 1, 7, 5, 9, 5, 3, 3, 3, 7, 7, 6

Offset: 1

Paul D. Hanna, Dec 03 2015

			z = 0.39318847704964432449725821313890388585483914078866\
28695392932475757877583389749866810976662061018588\
80133300805932263153268090475049426661211424334984\
43584775850655933725091432887705432231407717175953\
33776901692614854937460993931094741172922114373160\
19617637538747813543456758934332723336245738884968...
INFINITE SERIES.
(1) z = 1/4 + 1/4^2 + 3/4^3 + 1/4^4 + 19/4^5 + 25/4^6 + 11/4^8 + 161/4^9 + 227/4^10 + 681/4^11 + 1019/4^12 +...+ A002380(n)/4^n +...
(2) 3 - z = 1/2 + 2/2^2 + 3/2^3 + 5/2^4 + 7/2^5 + 11/2^6 + 17/2^7 + 25/2^8 + 38/2^9 + 57/2^10 + 86/2^11 + 129/2^12 + 194/2^13 + 291/2^14 +...+ A002379(n)/2^n +...
where
3 - z = 2.60681152295035567550274178686109611414516...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A002379 ([(3/2)^n]), A002380 (3^n mod 2^n), A264919, A264920, A264921, A264922.

z = Sum_{n>=1} (3^n mod 2^n) / 4^n = Sum_{n>=1} A002380(n) / 4^n.

3 - z = Sum_{n>=1} [(3/2)^n] / 2^n = Sum_{n>=1} A002379(n) / 2^n, where [x] denotes the integer floor function of x.

cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Extensions

A060692 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A065554 Numbers k such that floor((3/2)^(k+1))/floor((3/2)^k) = 3/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A138589 a(n) = 5^n mod 4^n.

Views

Author

Keywords

Links

Crossrefs

Programs