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.

Showing 1-10 of 11 results. Next

A085276 Integer reached in A072340.

Original entry on oeis.org

1, 8, 1484710602474311520, 2, 7, 8, 3, 1484710602474311520, 220, 4, 227370, 560, 5, 32, 34, 6, 1995, 23157015398160, 7, 11983136, 209136187376736, 8, 75, 78, 9
Offset: 3

Views

Author

N. J. A. Sloane, Aug 13 2003

Keywords

References

  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

Crossrefs

Cf. A072340, A085285, A085286.

Extensions

The next term is too large to include.

A085285 Positions where records occur in A072340.

Original entry on oeis.org

3, 4, 5, 28, 1783, 7148, 273223, 398314, 1180939, 1751431, 10970993, 17545207, 20110862, 190270082, 10261454491
Offset: 1

Views

Author

N. J. A. Sloane, Aug 13 2003

Keywords

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A072340, A085276, A085286.

Extensions

More terms from Vim Wenders, May 05 2008

A085286 Record values in A072340.

Original entry on oeis.org

0, 2, 6, 22, 23
Offset: 0

Views

Author

N. J. A. Sloane, Aug 13 2003

Keywords

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A072340, A085276, A085285.

A073524 Number of steps to reach an integer starting with (n+1)/n and using the map x -> x*ceiling(x); or -1 if no integer is ever reached.

Original entry on oeis.org

0, 1, 2, 3, 18, 2, 3, 4, 6, 7, 26, 4, 9, 3, 4, 8, 6, 4, 56, 11, 3, 4, 42, 4, 33, 7, 5, 4, 38, 5, 79, 6, 4, 15, 14, 8, 200, 29, 13, 5, 36, 3, 4, 5, 7, 10, 11, 8, 6, 20, 47, 27, 43, 9, 41, 9, 10, 23, 37, 17, 18, 6, 7, 6, 32, 15, 225, 7, 73, 11, 20, 12, 182, 9, 16, 7, 10, 15, 196, 8
Offset: 1

Views

Author

N. J. A. Sloane, Aug 29 2002

Keywords

Comments

Computed by doing all computations over the integers (multiply by n) and by truncating modulo n^250. This avoids the explosion of the integers (of order 2^(2^k) after k iterations) and gives the correct answer if the final index i(n) is < 250 (or perhaps 249 or 248). If the algorithm does not stop before 245 one should increase precision (work with n^500 or even higher). - Roland Bacher
Always reaches an integer for n <= 100. - Roland Bacher, Aug 30 2002
Always reaches an integer for n <= 200. - N. J. A. Sloane, Sep 04 2002
Always reaches an integer for n <= 500 by comparing results with index 1000 and index 2500. - Robert G. Wilson v, Sep 11 2002
Always reaches an integer for n <= 3000. The Mathematica program automatically adjusts the modulus m required to find the first integral iterate. - T. D. Noe, Apr 10 2006
Always reaches an integer for n <= 5000. - Ben Branman, Feb 12 2011

Examples

			a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.

Links

Crossrefs

Cf. A073528, A073529, A068119, A001511, A072340, A075102, A073341.

Programs

  • Mathematica
    Table[{n, First[Flatten[Position[Map[Denominator, NestList[ # Ceiling[ # ] &, (n + 1)/n, 20]], 1]]]}, {n, 1, 20}]
f[n_] := Block[{k = (n + 1)/n, c = 0}, While[ !IntegerQ[k], c++; k = Mod[k*Ceiling[k], n^250]]; c]; Table[ f[n], {n, 1, 100}]
Table[lim=50; While[k=0; x=1+1/n; m=n^lim; While[kT. D. Noe, Apr 10 2006 *)

Extensions

a(5)-a(10), a(12)-a(18), a(20) = 11 from Ed Pegg Jr, Aug 29 2002
T. D. Noe also found a(5) and remarks that the final integer is 9.5329600...*10^57734. - Aug 29 2002
a(11) from T. D. Noe, who remarks that the final integer is 5.131986636061311...*10^13941166 - Aug 29 2002
a(19) and a(21) onwards from Roland Bacher, Aug 30 2002

A074078 Number of steps to reach an integer starting with s = n/3 and iterating the map x -> s*ceiling(x).

Original entry on oeis.org

0, 2, 4, 0, 1, 1, 0, 13, 2, 0, 3, 2, 0, 1, 1, 0, 2, 4, 0, 8, 5, 0, 1, 1, 0, 7, 9, 0, 2, 7, 0, 1, 1, 0, 3, 2, 0, 6, 2, 0, 1, 1, 0, 2, 3, 0, 10, 3, 0, 1, 1, 0, 3, 3, 0, 2, 3, 0, 1, 1, 0, 5, 2, 0, 5, 2, 0, 1, 1, 0, 2, 10, 0, 3, 7, 0, 1, 1, 0, 8, 4, 0, 2, 6, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 1, 1, 0, 2, 5, 0, 4, 6, 0, 1, 1, 0
Offset: 3

Views

Author

N. J. A. Sloane, Sep 16 2002

Keywords

Examples

			s = 5/3 -> 10/3 -> 20/3 -> 35/3 -> 20, so a(5) = 4.

Links

Crossrefs

Cf. A072340, A074090, A074091, A074096. Records are in A074097, A074098.
First integer reached: A081852.

Programs

  • Maple
    f := proc(b1,b2) local c1,c2,t1,t2,t3,t4,i; c1 := numer(b1/b2); c2 := denom(b1/b2); i := 0; while c2 <> 1 do i := i+1; t1 := ceil(c1/c2); t2 := b1*t1; t3 := t2/b2; c1 := numer(t3); c2 := denom(t3); od: RETURN(i); end; [seq(f(n,3),n=4..120)];
  • Mathematica
    ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; Table[ce[n], {n, 3, 108}] (* Jayanta Basu, Jul 30 2013 *)
  • Python
    from math import ceil
from fractions import Fraction
def a(n):
    s = Fraction(n, 3)
    x, c = s, 0
    while x.denominator != 1:
        U = ceil(x)
        x, c = U*s, c+1
    return c
print([a(n) for n in range(3, 109)]) # Michael S. Branicky, Jan 09 2025

A117596 Start with x=6/5; repeatedly apply the map x -> x*ceiling(x); sequence gives numerators of the resulting sequence of fractions.

Original entry on oeis.org

6, 12, 36, 288, 16704, 55808064, 622908012647232, 77602878444025201997703040704, 1204441348559630271252918141028336694332989128001036771264, 290135792424028156178425357986052529062710984863337179470336908191924417208517059859206222048920739921330978585792
Offset: 1

Views

Author

N. J. A. Sloane, Apr 07 2006

Keywords

Comments

After 18 terms the fractions become integers, the first of which has 57735 digits.

Examples

			The sequence of fractions begins 6/5, 12/5, 36/5, 288/5, 16704/5, 55808064/5, 622908012647232/5, 77602878444025201997703040704/5, ... The first 17 denominators are 5, the rest are 1.

References

  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

Crossrefs

Cf. A072340, A085276.

Programs

  • Mathematica
    f[x_] := x*Ceiling[x]; NestList[f, 6/5, 9] // Numerator (* Jean-François Alcover, Nov 18 2013 *)

A117620 Start with x=4/3; repeatedly apply the map x -> (x^2) ceiling(x); sequence gives numerators of the resulting sequence of fractions.

Original entry on oeis.org

4, 32, 4096, 285212672, 3536203627938199896064, 27735467127437590594631628902073909856749798039036448735232
Offset: 1

Views

Author

Jonathan Vos Post, Apr 07 2006

Keywords

Comments

In this approximate cubing, does an iteration eventually yield an integer, after which denominators are 1? Fractions are 4/3, 32/9, 4096/81, 285212672/2187, 3536203627938199896064/1594323, 27735467127437590594631628902073909856749798039036448735232/2541865828329, 8393707510592229745861012598171776416393703955772365464679357805492895042198412632866136478758067686243059846017657263750451410617880163800261945260539460460740608/6461081889226673298932241.
a(9) has 1343 digits, and is too large for a b-file. - Robert Israel, Jun 15 2016

Examples

			a(4) = 285212672 because (4096/81)^2 * ceiling(4096/81) = (4096/81)^2 * ceiling(4096/81) = * ceiling(50.5679012) = (16777216/6561) * 51 = 285212672/2187.

Links

Crossrefs

Cf. A072340, A085276, A117596.

Programs

  • Maple
    x[1]:= 4/3:
for n from 1 to 9 do x[n+1]:= x[n]^2*ceil(x[n]) od:
seq(numer(x[i]),i=1..10); # Robert Israel, Jun 15 2016

Extensions

Erroneous term removed by Giovanni Resta, Jun 15 2016

A074090 Number of steps to reach an integer starting with s = n/4 and iterating the map x -> s*ceiling(x).

Original entry on oeis.org

3, 1, 2, 0, 3, 2, 13, 0, 1, 1, 1, 0, 4, 3, 2, 0, 2, 1, 9, 0, 2, 2, 2, 0, 1, 1, 1, 0, 7, 4, 3, 0, 7, 1, 3, 0, 4, 2, 3, 0, 1, 1, 1, 0, 2, 3, 4, 0, 5, 1, 10, 0, 5, 2, 13, 0, 1, 1, 1, 0, 14, 5, 8, 0, 5, 1, 2, 0, 10, 2, 11, 0, 1, 1, 1, 0, 6, 3, 2, 0, 2, 1, 17, 0, 2, 2, 2, 0, 1, 1, 1, 0, 3, 4, 6, 0, 16, 1, 5, 0, 3, 2, 4
Offset: 5

Views

Author

N. J. A. Sloane, Sep 16 2002

Keywords

Crossrefs

Cf. A072340, A074078.

Programs

  • Maple
    f := proc(b1,b2) local c1,c2,t1,t2,t3,t4,i; c1 := numer(b1/b2); c2 := denom(b1/b2); i := 0; while c2 <> 1 do i := i+1; t1 := ceil(c1/c2); t2 := b1*t1; t3 := t2/b2; c1 := numer(t3); c2 := denom(t3); od: RETURN(i); end; [seq(f(n,4),n=5..120)];

A074091 Number of steps to reach an integer starting with s = n/5 and iterating the map x -> s*ceiling(x).

Original entry on oeis.org

4, 3, 5, 4, 0, 5, 3, 4, 5, 0, 4, 9, 2, 8, 0, 1, 1, 1, 1, 0, 7, 7, 4, 2, 0, 14, 2, 5, 7, 0, 4, 2, 9, 10, 0, 5, 8, 10, 2, 0, 1, 1, 1, 1, 0, 7, 2, 4, 17, 0, 2, 5, 2, 6, 0, 3, 17, 4, 6, 0, 2, 3, 6, 11, 0, 1, 1, 1, 1, 0, 11, 7, 2, 12, 0, 12, 6, 5, 3, 0, 2, 9, 3, 4, 0, 8, 2, 3, 6, 0, 1, 1, 1, 1, 0, 2, 6, 3, 3, 0, 10
Offset: 6

Views

Author

N. J. A. Sloane, Sep 16 2002

Keywords

Crossrefs

Cf. A072340, A074078.

A074096 Number of steps to reach an integer starting with s = n/7 and iterating the map x -> s*ceiling(x).

Original entry on oeis.org

6, 25, 16, 3, 3, 5, 0, 2, 2, 4, 3, 12, 10, 0, 5, 2, 2, 11, 3, 8, 0, 2, 8, 5, 23, 13, 11, 0, 6, 21, 6, 5, 2, 6, 0, 1, 1, 1, 1, 1, 1, 0, 21, 5, 5, 3, 4, 2, 0, 6, 12, 6, 7, 14, 7, 0, 13, 15, 3, 19, 2, 5, 0, 2, 24, 7, 5, 5, 4, 0, 5, 20, 5, 19, 3, 5, 0, 4, 12, 4, 15, 7, 2, 0, 1, 1, 1, 1, 1, 1, 0, 14, 17, 2, 3
Offset: 8

Views

Author

N. J. A. Sloane, Sep 16 2002

Keywords

Crossrefs

Cf. A072340, A074078.
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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