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

A074069 Numbers k such that S(k) = Sum_{i=1..k} A068119(i) = 3*k.

Original entry on oeis.org

1, 2, 10, 18, 20, 25, 26, 43, 46, 55, 56, 71, 79, 104, 107, 111, 112, 113, 123, 133, 136, 140, 142, 144, 145, 147, 150, 155, 165, 195, 196, 200, 204, 206, 210, 214, 216, 253, 254, 259, 260, 264, 267, 327, 335, 343, 360, 363, 375, 376, 380, 387, 388, 393
Offset: 1

Views

Author

Benoit Cloitre, Aug 30 2002

Keywords

Programs

  • Mathematica
    ce[n_] := Length[NestWhileList[#*Ceiling[#] &, n + 1/4, ! IntegerQ[#] &]] - 1; nn = Range[395]; t = Accumulate[ce /@ nn]; Select[nn, t[[#]] == 3*# &] (* Jayanta Basu, Jul 29 2013 *)

Extensions

Name clarified by Sean A. Irvine, Jan 08 2025

A074077 a(n) = 3n - Sum_{k=1..n} A068119(n).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Aug 30 2002

Keywords

Programs

  • Mathematica
    ce[n_] := Length[NestWhileList[#*Ceiling[#] &, n + 1/4, ! IntegerQ[#] &]] - 1; nn = Range[83]; t = Accumulate[ce /@ nn]; Table[3 n - t[[n]], {n, nn}] (* Jayanta Basu, Jul 30 2013 *)

Formula

a(n) = a(n-1) + 3 - A068119(n) with a(0)=0. - Sean A. Irvine, Jan 08 2025

Extensions

Name clarified by Sean A. Irvine, Jan 08 2025

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

A075102 Number of steps to reach the first integer starting with 2^n/n and iterating the map x->x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 2, 5, 5, 0, 2, 4, 1, 1, 13, 12, 3, 0, 19, 3, 4, 8, 10, 11, 20, 1, 7, 1, 3, 8, 12, 1, 2, 0, 20, 6, 4, 7, 8, 23, 39, 1, 21, 8, 13, 16, 4, 1, 9, 7, 1, 6, 23, 30, 73, 6, 3, 14, 7, 7, 20, 12, 228, 16, 3, 0, 10, 5, 96, 3, 4, 13, 28, 3, 72, 57, 7, 9, 6, 6
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 02 2002

Keywords

Comments

The starting value is A075101(n)/A000265(n).

Examples

			a(5)=2 since (2^5)/5 = 32/5 -> 224/5 -> 2016 = A075103(5).

Links

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

Crossrefs

Cf. A000265, A068119, A073524, A075101, A075103.

Extensions

More terms from Jinyuan Wang, Jan 15 2022

A075107 Number of steps to reach the first integer (= A075108(n)) starting with n/floor(log_2(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 0, 3, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 2, 5, 3, 0, 16, 2, 4, 6, 0, 4, 16, 9, 2, 0, 1, 1, 1, 1, 0, 4, 2, 4, 8, 0, 2, 5, 2, 8, 0, 9, 3, 4, 5, 2, 0, 1, 1, 1, 1, 1, 0, 5, 7, 3, 2, 4, 0, 9, 4, 1, 2, 3, 0, 4, 1, 2, 1
Offset: 2

Views

Author

Reinhard Zumkeller, Sep 02 2002

Keywords

Comments

Starting values given by A075105(n)/A075106(n).

Examples

			a(5)=2 since 5/floor(log_2(5)) = 5/2 -> 15/2 -> 60 = A075108(5).

Crossrefs

Cf. A068119, A073524, A075102, A075120.

Extensions

More terms from Jinyuan Wang, Jan 15 2022

A075120 Number of steps to reach the first integer (= A075121(n)) starting with n/floor(sqrt(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 0, 0, 5, 2, 0, 3, 2, 0, 0, 3, 3, 2, 0, 2, 1, 3, 0, 0, 8, 3, 5, 2, 0, 3, 2, 5, 3, 0, 0, 6, 2, 2, 4, 2, 0, 5, 3, 1, 4, 2, 0, 0, 7, 12, 5, 11, 12, 2, 0, 4, 3, 9, 12, 6, 16, 0, 0, 3, 7, 8, 4, 4, 4, 2, 0, 2, 4, 5, 1, 5, 2, 5, 0, 0, 7, 3, 22, 3, 3
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 03 2002

Keywords

Comments

Starting with A073890(n)/A075119(n).

Examples

			a(13)=3 since 13/floor(sqrt(13)) = 13/3 -> 65/3 -> 1430/3 -> 227370 = A075121(13).

Crossrefs

Cf. A068119, A073524, A075102, A075107, A075121.

Extensions

More terms from Jinyuan Wang, Jan 15 2022

A074735 Number of steps to reach an integer starting with (n+3)/4 and iterating the map x -> x*ceiling(x).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Sep 05 2002

Keywords

Comments

Let S(n) = Sum_{k=1..n} a(k) then it seems that S(n) is asymptotic to 2n. S(n)=2n for many values of n, namely n=10,128,198,199,237,238,241,242,246,247,249,267,329... More generally, starting with (n+2^m-1)/2^m and iterating the same map seems to produce the same kind of behavior for a(n) (i.e., Sum_{k=1..n} a(k) is asymptotic to c(m)*n where c(m) depends on m and c(m) is a power of 2).

Crossrefs

Cf. A073524, A073341, A068119.

Programs

  • Mathematica
    Table[Length[NestWhileList[# Ceiling[#]&,(n+3)/4,!IntegerQ[#]&]]-1,{n,110}] (* Harvey P. Dale, Apr 11 2020 *)
  • PARI
    a(n)=if(n<0,0,s=(n+3)/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Formula

Special cases: for k>= 0 a(4k+1) = 0, a(16k+10) = a(16k+11) = a(16k+12) = 1.

Extensions

Offset corrected by Sean A. Irvine, Jan 25 2025

A081882 Number of steps needed to reach an integer starting with n-1/4 and iterating the map x ->x*ceiling(x).

Original entry on oeis.org

2, 8, 1, 2, 3, 2, 1, 4, 2, 3, 1, 4, 8, 3, 1, 4, 2, 4, 1, 2, 3, 2, 1, 5, 2, 5, 1, 3, 4, 6, 1, 8, 2, 5, 1, 2, 3, 2, 1, 3, 2, 6, 1, 5, 5, 3, 1, 3, 2, 3, 1, 2, 3, 2, 1, 8, 2, 7, 1, 3, 4, 4, 1, 4, 2, 4, 1, 2, 3, 2, 1, 5, 2, 3, 1, 4, 6, 3, 1, 5, 2, 8, 1, 2, 3, 2, 1, 6, 2, 4, 1, 3, 4, 5, 1, 4, 2, 4, 1, 2, 3, 2, 1, 3, 2
Offset: 2

Views

Author

Benoit Cloitre, Apr 13 2003

Keywords

Crossrefs

Cf. A068119.

Programs

  • PARI
    a(n)=if(n<0,0,s=n-1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Formula

It seems that sum(k=2, n, a(k)) is asymptotic to 3n

A081883 Numbers n such that the number of steps needed to reach an integer starting with n-1/4 or n+1/4 and iterating the map x-->x*ceiling(x) is the same.

Original entry on oeis.org

5, 9, 21, 37, 46, 49, 53, 69, 85, 101, 113, 117, 121, 133, 137, 149, 165, 174, 177, 181, 197, 213, 217, 229, 241, 245, 261, 265, 277, 293, 302, 305, 309, 313, 325, 341, 357, 369, 373, 389, 393, 405, 421, 430, 433, 437, 441, 453, 469, 485, 497, 501, 517, 521
Offset: 1

Views

Author

Benoit Cloitre, Apr 13 2003

Keywords

Crossrefs

Cf. A068119.

A081884 Number of steps needed to reach an integer starting with n+1/8 and iterating the map x-->x*ceiling(x).

Original entry on oeis.org

4, 7, 4, 7, 7, 6, 1, 3, 2, 2, 2, 7, 2, 3, 1, 6, 5, 4, 3, 7, 7, 6, 1, 3, 3, 5, 2, 2, 3, 3, 1, 5, 3, 9, 6, 4, 7, 3, 1, 3, 2, 3, 2, 3, 2, 2, 1, 5, 5, 4, 3, 10, 3, 4, 1, 2, 4, 5, 2, 7, 7, 9, 1, 6, 6, 3, 4, 12, 3, 13, 1, 5, 2, 2, 2, 10, 2, 5, 1, 5, 3, 13, 3, 3, 5, 6, 1, 9, 3, 6, 2, 2, 3, 4, 1, 6, 6, 6, 5, 6, 4, 4, 1
Offset: 1

Views

Author

Benoit Cloitre, Apr 13 2003

Keywords

Comments

Conjecture : let b(n,m) denotes the number of steps needed to reach an integer starting with n+1/2^m and iterating the map x-->x*ceiling(x); then sum(k=1,n,b(k,m)) is asymptotic to (m+1)*n.

Crossrefs

Cf. A068119, A073524.

Programs

  • PARI
    a(n)=if(n<0,0,s=n+1/8; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Formula

It seems that sum(k=1, n, a(k)) is asymptotic to 4n.
Showing 1-10 of 11 results. Next

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