A002379 -id:A002379 - cp's OEIS frontend

A153663 Minimal exponents m such that the fractional part of (3/2)^m reaches a maximum (when starting with m=1).

1, 5, 8, 10, 12, 14, 46, 58, 105, 157, 163, 455, 1060, 1256, 2677, 8093, 28277, 33327, 49304, 158643, 164000, 835999, 2242294, 25380333, 92600006

Offset: 1

Hieronymus Fischer, Dec 31 2008

Recursive definition: a(1)=1, a(n) = least number m such that the fractional part of (3/2)^m is greater than the

fractional part of (3/2)^k for all k, 1<=k

The fractional part of k=835999 is .999999 5 which is greater than (k-1)/k. The fractional part of k=2242294 is .999999 8 which is greater than (k-1)/k. The fractional part of k=25380333 is .999999 98 which is greater than (k-1)/k. The fractional part of k=92600006 is .999999 998 which is greater than (k-1)/k. So, all additional numbers in this sequence must be in A153664 and >3*10^8. - Robert Price, May 09 2012

			a(2)=5, since fract((3/2)^5)=0.59375, but fract((3/2)^k)=0.5, 0.25, 0.375, 0.0625 for 1<=k<=4; thus
fract((3/2)^5)>fract((3/2)^k) for 1<=k<5.

Cf. A002379, A153661, A153662, A153664, A153665, A153666, A153667, A153668.

a[1] = 1; a[n_] := a[n] = For[m = a[n-1]+1, True, m++, f = FractionalPart[(3/2)^m]; If[AllTrue[Range[m-1], f > FractionalPart[(3/2)^#]&], Print[n, " ", m]; Return[m]]];
Array[a, 21] (* Jean-François Alcover, Feb 25 2019 *)

Recursion: a(1):=1, a(k):=min{ m>1 | fract((3/2)^m) > fract((3/2)^a(k-1))}, where fract(x) = x-floor(x).

a(22)-a(25) from Robert Price, May 09 2012

A153664 Numbers k such that the fractional part of (3/2)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 14, 163, 1256, 2677, 8093, 49304, 49305, 158643, 164000, 835999, 2242294, 2242295, 2242296, 3965133, 25380333, 92600006, 92600007, 92600008, 92600009, 92600010, 92600011, 99267816, 125040717, 125040718

Offset: 1

Hieronymus Fischer, Dec 31 2008

Numbers k such that fract((3/2)^k) > 1-(1/k), where fract(x) = x-floor(x).

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

			a(2) = 14 since fract((3/2)^14) = 0.92926... > 0.92857... = 1 - (1/14), but fract((3/2)^k) <= 1 - (1/k) for 1<k<14.

Cf. A002379, A081464, A153662, A153663, A153665, A153666, A153667, A153668.

Select[Range[1000], FractionalPart[(3/2)^#] >= 1 - (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

a(11)-a(25) from Robert Gerbicz, Nov 21 2010

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

A153665 Greatest number m such that the fractional part of (3/2)^A081464(n) <= 1/m.

Original entry on oeis.org

2, 4, 16, 25, 89, 91, 105, 127, 290, 668, 869, 16799, 92694, 137921, 257825, 350408, 419427, 723749, 5271294, 14223700, 18090494, 88123482, 706641581

Offset: 1

Hieronymus Fischer, Dec 31 2008

			a(4)=25 since 1/26<fract((3/2)^A081464(4))=fract((3/2)^29)=0.039...<=1/25.

Cf. A002379, A153662, A153663, A153664, A153666, A153667, A153668.

A081464 = {1, 2, 4, 29, 95, 153, 532, 613, 840, 2033, 2071, 3328, 12429, 112896, 129638, 371162, 1095666, 3890691, 4264691, 31685458, 61365215, 92432200, 144941960};
Table[fp = FractionalPart[(3/2)^A081464[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++];  m - 1, {n, 1, Length[A081464]}] (* Robert Price, Mar 26 2019 *)

a(n):=floor(1/fract((3/2)^A081464(n))), where fract(x) = x-floor(x).

a(16)-a(23) from Robert Price, May 09 2012

A153666 Greatest number m such that the fractional part of (3/2)^A153662(n) <= 1/m.

Original entry on oeis.org

2, 4, 16, 11, 16799, 11199, 5536, 92694, 61796, 41197, 23242, 55710, 137921, 257825, 5271294, 706641581, 471094387, 314062925

Offset: 1

Hieronymus Fischer, Dec 31 2008

			a(3)=16 since 1/17<fract((3/2)^A153662(3))=fract((3/2)^4)=0.0625=1/16.

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153667, A153668.

A153662 = {1, 2, 4, 7, 3328, 3329, 4097, 12429, 12430, 12431, 18587, 44257, 112896, 129638, 4264691, 144941960, 144941961, 144941962};
Table[fp = FractionalPart[(3/2)^A153662[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++];  m - 1, {n, 1, Length[A153662]}] (* Robert Price, Mar 26 2019 *)

a(n):=floor(1/fract((3/2)^A153662(n))), where fract(x) = x-floor(x).

a(15)-a(18) from Robert Price, May 09 2012

A153667 Greatest number m such that the fractional part of (3/2)^A153663(n) >= 1-(1/m).

Original entry on oeis.org

2, 2, 2, 2, 3, 14, 31, 33, 69, 137, 222, 318, 901, 1772, 2747, 12347, 16540, 18198, 135794, 222246, 570361, 2134829, 6901329, 75503109, 814558605

Offset: 1

Hieronymus Fischer, Dec 31 2008

			a(5)=3, since 1-(1/4)=0.75>fract((3/2)^A153663(5))=fract((3/2)^12)=0.746...>=1-(1/3).

Cf. A002379, A081464, A153662, A153663, A153664, A153665, A153666, A153668.

A153663 = {1, 5, 8, 10, 12, 14, 46, 58, 105, 157, 163, 455, 1060, 1256, 2677, 8093, 28277, 33327, 49304, 158643, 164000, 835999, 2242294, 25380333, 92600006};
Table[fp = FractionalPart[(3/2)^A153663[[n]]]; m = Floor[1/(1-fp)];
While[fp >= 1 - (1/m), m++]; m - 1, {n, 1, Length[A153663]}] (* Robert Price, Mar 26 2019 *)

a(n) = floor(1/(1-fract((3/2)^A153663(n)))), where fract(x) = x-floor(x).

a(22)-a(25) from Robert Price, May 10 2012

Previous Showing 11-20 of 90 results. Next

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

cp's OEIS Frontend

A153662 Numbers k such that the fractional part of (3/2)^k is less than 1/k.

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

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A153663 Minimal exponents m such that the fractional part of (3/2)^m reaches a maximum (when starting with m=1).

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A153664 Numbers k such that the fractional part of (3/2)^k is greater than 1-(1/k).

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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.

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

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

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