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 1-8 of 8 results.

A074097 Record values in A074078.

Original entry on oeis.org

2, 4, 13, 15, 21, 22, 23, 27, 29, 34, 36, 40, 42, 50, 75, 81
Offset: 1

Views

Author

N. J. A. Sloane, Sep 17 2002

Keywords

Comments

RECORDS transform of A074078.

Links

Crossrefs

Cf. A074078, A074098.

Programs

  • Mathematica
    ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; t = {}; r = 0; Do[If[ce[n] > r, AppendTo[t, r = ce[n]]], {n, 3, 5*10^5}]; t (* Jayanta Basu, Jul 31 2013 *)

Extensions

a(13)-a(14) from Sean A. Irvine, Jan 09 2025
a(15)-a(16) from Michael S. Branicky, Jan 09 2025

A074098 Where records occur in A074078.

Original entry on oeis.org

4, 5, 10, 275, 508, 3694, 5675, 6562, 32420, 137782, 295409, 430645, 13564046, 22289981, 387420490, 3486784402
Offset: 1

Views

Author

N. J. A. Sloane, Sep 17 2002

Keywords

Comments

RECORDS transform of A074078.

Links

Crossrefs

Cf. A074078, A074097.

Programs

  • Mathematica
    ce[n_] := Length[NestWhileList[(n/3)*Ceiling[#] &, n/3, ! IntegerQ[#] &]] - 1; t = {}; r = 0; Do[If[ce[n] > r, AppendTo[t, n]; r = ce[n]], {n, 3, 5*10^5}]; t (* Jayanta Basu, Jul 31 2013 *)

Extensions

a(9)-a(12) from Jayanta Basu, Jul 31 2013
a(13)-a(14) from Sean A. Irvine, Jan 09 2025
a(15)-a(16) from Michael S. Branicky, Jan 09 2025

A081849 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

3, 20, 14, 468, 33, 299, 60, 47328, 95, 1218, 138, 25475, 189, 3161, 248, 20830128, 315, 6512, 390, 181138, 473, 11655, 564, 9015167, 663, 18974, 770, 671745, 885, 28853, 1008, 38906570560, 1139, 41676, 1278, 1799888, 1425, 57827, 1580, 110341278, 1743, 77690
Offset: 1

Views

Author

N. J. A. Sloane, Apr 13 2003

Keywords

Comments

k = A001511(n) is the number of steps to reach an integer b(k).

Crossrefs

Cf. A001511, A081853, A073524, A074078.

Programs

  • Maple
    Digits := 100: c := ceil; A081849 := proc(a) local i,t0,t; t0 := a; t := 0; for i from 1 to 100 do if whattype(t0) <> integer then t0 := a*c(t0); t := t+1; else RETURN(t0); fi; od; RETURN('FAIL'); end;
  • Mathematica
    a[n_]:=Module[{b=b0=(2n+1)/2},While[!IntegerQ[b],b=b0*Ceiling[b]]; b]; Array[a,42] (* Stefano Spezia, Jun 26 2024 *)
  • PARI
    a(n) = if(n==1,3, my(t=2*n+1, k=1+valuation(n,2)); n*t^(k+1) >>k \ (t-2)); \\ Kevin Ryde, Jun 30 2024
  • Python
    from math import ceil
from fractions import Fraction
def a(n):
  b0 = b = Fraction((2*n+1), 2)
  while b.denominator != 1: b = b0*ceil(b)
  return b.numerator
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Mar 20 2021

Formula

a(n) = s*(n*s^k - 1/2) / (s-1) where s = b(0) = (2*n+1)/2 and k = A001511(n). - Kevin Ryde, Jun 30 2024

A081853 Consider recurrence b(0) = (2n+1)/2, b(n) = b(n-1)*ceiling(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

3, 60, 14, 268065, 33, 2093, 60, 1204154941925628, 95, 13398, 138, 701600900, 189, 47415, 248, 1489788110004539889867929328515560588293, 315, 123728, 390, 34427225343, 473, 268065, 564, 19873182780430314444725, 663, 512298, 770, 467193780498, 885, 894443, 1008
Offset: 1

Views

Author

N. J. A. Sloane, Apr 13 2003

Keywords

Links

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

Crossrefs

A001511 gives number of steps to reach an integer. Cf. A081849, A073524, A074078.
Cf. A081854.

Programs

  • Mathematica
    a[n_]:=Module[{b=(2n+1)/2},While[!IntegerQ[b],b*=Ceiling[b]]; b]; Array[a,31] (* Stefano Spezia, Jun 26 2024 *)

Formula

Define F(x) = x(x+1)/2. Write 2n+1 = 2^i*m + 2^(i-1) + 1, then a(n) = (1/2)F^(i-1)(2n+1). E.g. n=4, 2n+1 = 9 = 2^4*0 + 2^3 + 1, so i=4, m=0 and F(F(F(9))) = F(F(45)) = F(1035) = 536130, a(4) = 536130/2 = 268065.

Extensions

a(30)-a(31) from Stefano Spezia, Jun 26 2024

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.

A081852 Consider recurrence b(0) = n/3, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached (or -1 if no integer is ever reached).

Original entry on oeis.org

1, 4, 20, 2, 7, 8, 3, 26631380, 55, 4, 416, 112, 5, 32, 34, 6, 285, 13960, 7, 67358874, 214544, 8, 75, 78, 9, 62186796, 7399041846, 10, 1178, 173857344, 11, 136, 140, 12, 24494, 2090, 13, 78824360, 2624, 14, 215, 220, 15, 3772, 61617, 16, 23001295794169, 78900, 17, 312
Offset: 3

Views

Author

N. J. A. Sloane, Apr 13 2003

Keywords

Crossrefs

Cf. A073524, A081849, A081850, A081851. A074078 gives number of steps to reach an integer.
Showing 1-8 of 8 results.

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

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