cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

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

Views

Author

Labos Elemer, Oct 08 2001

Keywords

Comments

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.

Links

Crossrefs

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

Programs

Formula

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

A070981 Smallest integer k such that n divides floor((4/3)^k).

Original entry on oeis.org

1, 3, 4, 5, 6, 13, 7, 14, 8, 21, 16, 21, 9, 13, 20, 22, 10, 32, 17, 21, 13, 51, 11, 67, 37, 66, 65, 14, 69, 21, 12, 68, 16, 35, 20, 66, 15, 122, 65, 22, 98, 13, 70, 66, 20, 117, 28, 67, 58, 37, 34, 66, 151, 103, 93, 14, 240, 80, 18, 21, 79, 87, 20, 68, 114, 66, 28, 35, 155
Offset: 1

Views

Author

Benoit Cloitre, May 24 2002

Keywords

Links

Crossrefs

Cf. A064628.

Programs

  • Mathematica
    sik[n_]:=Module[{k=1},While[Mod[Floor[(4/3)^k],n]!=0,k++];k]; Array[sik,70] (* Harvey P. Dale, Jun 12 2019 *)
  • PARI
    for(n=1,100,s=1; while(floor((4/3)^s)%n>0,s++); print1(s,","))

Formula

a(n) = min( k : A064628(k) == 0 mod(n) )

A154131 Numbers n such that the fractional part of (4/3)^n is less than 1/n.

Original entry on oeis.org

1, 4, 17, 1738, 1739, 12863, 15705, 109705, 174894, 289047, 720429, 2087694, 2087695, 4475944, 6968999
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Comments

Numbers n such that fract((4/3)^n) < 1/n, where fract(x) = x - floor(x).
The next term is greater than 3*10^8.

Examples

			a(3)=17 since fract((4/3)^17) = 0.03273... < 1/17, but fract((4/3)^k) >= 1/k for 5 <= k <= 16.

Crossrefs

Cf. A153662, A153670, A153702, A137994, A154139, A154147.
Cf. A002379, A064628.

Programs

  • Mathematica
    Select[Range[1000], N[FractionalPart[(4/3)^#], 100] < (1/#) &] (* G. C. Greubel, Sep 02 2016 *)
  • PARI
    isok(n) = frac((4/3)^n) < 1/n; \\ Michel Marcus, Sep 03 2016

Extensions

a(10)-a(15) from Robert Gerbicz, Nov 21 2010

A154132 Minimal exponents m such that the fractional part of (4/3)^m increases monotonically (when starting with m=1).

Original entry on oeis.org

1, 2, 8, 39, 2495, 3895, 4714, 8592
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Comments

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (4/3)^m is greater than the fractional part of (4/3)^k for all k, 1<=k
The next such number must be greater than 200000.

Examples

			a(4)=39, since fract((4/3)^39)= 0.999186..., but fract((4/3)^k)<0.9887... for 1<=k<=38; thus fract((4/3)^39)>fract((4/3)^k) for 1<=k<39 and 39 is the minimal exponent > 8 with this property.

Crossrefs

Cf. A153663, A153671, A091560, A153710, A154140, A154148.
Cf. A002379, A064628.

Formula

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).

A154133 Numbers k such that the fractional part of (4/3)^k is greater than 1-(1/k).

Original entry on oeis.org

1, 2, 8, 39, 113, 3895, 4714, 8592, 34289, 60097, 942859, 2759790, 6649343, 7937397, 14480816, 19338413, 19338414, 19338415, 23187701, 124679421
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Comments

Numbers k such that fract((4/3)^k) > 1-(1/k), where fract(x) = x - floor(x).
The next term is greater than 3*10^8.

Examples

			a(4) = 39, since fract((4/3)^39) = 0.9991861450... > 0.974358... = 1 - (1/39), but fract((4/3)^k) <= 1 - (1/k) for 8 < k < 39.

Crossrefs

Cf. A153664, A153672, A153704, A153711, A154141, A154149.
Cf. A002379, A064628.

Programs

  • Mathematica
    Select[Range[5000], N[FractionalPart[(4/3)^#], 100] > 1 - (1/#) &] (* G. C. Greubel, Sep 02 2016 *)
  • PARI
    isok(n) = frac((4/3)^n) > 1 - 1/n; \\ Michel Marcus, Sep 03 2016

Extensions

a(11)-a(20) from Robert Gerbicz, Nov 21 2010

A179492 a(n) = floor((1 + 1/Pi)^n).

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 9, 12, 15, 20, 27, 36, 47, 63, 83, 109, 144, 190, 251, 331, 436, 575, 759, 1001, 1319, 1739, 2293, 3023, 3985, 5254, 6927, 9132, 12039, 15871, 20923, 27583, 36363, 47938, 63197, 83313, 109833, 144794, 190884, 251644, 331745
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 16 2010

Keywords

Crossrefs

Cf. A002379 (with 2 instead of Pi), A064628 (with 3 instead of Pi).

Programs

  • Maple
    Digits := 100 ; A179492 := proc(n) (1+1/Pi)^n ; floor(%) ; end proc: seq(A179492(n),n=1..80) ; # R. J. Mathar, Jul 20 2010
  • Mathematica
    a=1; Table[Floor[a+=a/Pi],{n,0,22}]

Formula

a(n) = floor( (1+1/Pi)^n ). - R. J. Mathar, Jul 20 2010

Extensions

More terms from R. J. Mathar, Jul 20 2010

A154134 Greatest number m such that the fractional part of (4/3)^A154130(n) <= 1/m.

Original entry on oeis.org

3, 6, 10, 30, 124, 238, 405, 6430, 22869, 32657, 224544, 2396968, 15229280, 28274047, 53458049, 134537968
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=10 since 1/11<fract((4/3)^A154130(3))=fract((4/3)^13)=0.09238...<=1/10.

Crossrefs

Cf. A153665, A153673, A153705, A153712, A154143, A154150.
Cf. A002379, A064628.

Formula

a(n):=floor(1/fract((4/3)^A154130(n))), where fract(x) = x-floor(x).

Extensions

a(12)-a(16) from Robert Price, May 10 2012

A154135 Greatest number m such that the fractional part of (4/3)^A154131(n) <= 1/m.

Original entry on oeis.org

3, 6, 30, 6430, 4822, 22869, 20551, 224544, 184762, 2396968, 15229280, 3183837, 2387878, 28274047, 7149842
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=30 since 1/31<fract((4/3)^A154131(3))=fract((4/3)^17)=0.0327357...<=1/30.

Crossrefs

Cf. A153666, A153674, A153706, A153714, A154143, A154151.
Cf. A002379, A064628.

Formula

a(n) = floor(1/fract((4/3)^A154131(n))), where fract(x) = x - floor(x).

Extensions

a(10)-a(15) from Jinyuan Wang, Mar 03 2020

A154136 Greatest number m such that the fractional part of (4/3)^A154132(m) >= 1-(1/m).

Original entry on oeis.org

1, 4, 88, 1228, 2253, 4562, 8183, 167378
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=88, since 1-(1/89)=0.988764...>fract((4/3)^A154132(3))=fract((4/3)^8)=0.988721...>0.988636...=1-(1/88).

Crossrefs

Cf. A153667, A153675, A153707, A153715, A154144, A154152.
Cf. A002379, A064628.

Formula

a(n):=floor(1/(1-fract((4/3)^A154132(n)))), where fract(x) = x-floor(x).

A154137 Greatest number m such that the fractional part of (4/3)^A154133(n) >= 1-(1/m).

Original entry on oeis.org

1, 4, 88, 1228, 187, 4562, 8183, 167378, 35419, 77421, 5593723, 3306511, 83205705, 22413581, 24296709, 35457806, 26593355, 19945016, 80184972, 389460601
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Examples

			a(3)=88, since 1-(1/89)=0.988764...>fract((4/3)^A154133(3))=fract((4/3)^8)=0.988721...>0.988636...=1-(1/88).

Crossrefs

Cf. A153668, A153676, A153708, A153716, A154145, A154153.
Cf. A002379, A064628.

Formula

a(n):=floor(1/(1-fract((4/3)^A154133(n)))), where fract(x) = x-floor(x).

Extensions

a(11)-a(20) from Robert Price, May 10 2012
Previous Showing 11-20 of 20 results.

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