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 1-4 of 4 results.

A087226 LCM of terms in Collatz (3x+1) function initiated at n.

Original entry on oeis.org

1, 2, 240, 4, 80, 240, 1361360, 8, 12252240, 80, 194480, 240, 1040, 1361360, 4095840, 16, 17680, 12252240, 107158480, 80, 1344, 194480, 1365280, 240, 535792400, 1040, 44841486948146266934850832405421294927083491752830032389039800908293040266400
Offset: 1

Views

Author

Labos Elemer, Aug 28 2003

Keywords

Examples

			n=9: list={9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1};
LCM = 2*2*2*2*3*3*5*7*11*13*17 = 12252240.

Links

Crossrefs

Cf. A006370.
Cf. A070165, A225784.

Programs

  • Haskell
    a087226 = foldl1 lcm . a070165_row  -- Reinhard Zumkeller, May 16 2013
  • Mathematica
    c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] Table[Apply[LCM, fpl[w]], {w, 1, 32}]
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[LCM @@ Collatz[n], {n, 27}] (* T. D. Noe, May 15 2013 *)

A225761 Numerators of the sums of reciprocals of the Collatz (3x+1) sequence beginning with n and stopping at 1.

Original entry on oeis.org

1, 3, 617, 7, 171, 219, 766329, 15, 1061683, 179, 102151, 677, 497, 785777, 10380059, 31, 8861, 360377, 60226515, 183, 2731, 103919, 3339321, 229, 1548244271, 505, 129481899470258402665619129356105706380861444925035330406812603986229803685477, 113643
Offset: 1

Views

Author

Nico Brown, May 14 2013

Keywords

Comments

If the sum of the reciprocals of a Collatz sequence is bounded, there are no cycles other than 4,2,1.
a(n) = numerator of sum (1/A070165(n,k): k = 1..A006577(n)). - Reinhard Zumkeller, May 16 2013

Examples

			For n=9 the Collatz sequence is {9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 4, 2, 1}.  So the sum of the reciprocals is 1/9 + 1/28 + 1/14 + 1/7 + 1/22 + 1/11 + ... + 1/4 + 1/2 + 1/1 = 1061683/350064, whose numerator is 1061683.

Links

Crossrefs

Cf. A087226, A225784 (denominators).
Cf. A225843.

Programs

  • Haskell
    import Data.Ratio (numerator)
a225761 = numerator . sum . map (recip . fromIntegral) . a070165_row
-- Reinhard Zumkeller, May 16 2013
  • Mathematica
    Table[Numerator[Total[1/Collatz[n]]], {n, 40}] (* T. D. Noe, May 15 2013 *)

Extensions

Extended by T. D. Noe, May 15 2013

A225843 Integral parts of sums of the reciprocals of the Collatz (3x+1) sequence starting with n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3
Offset: 1

Views

Author

Reinhard Zumkeller, May 16 2013

Keywords

Comments

a(n) = floor(sum(1/A070165(n,k): k = 1..A006577(n)));
conjecture: a(n) <= 3;
a(n) = 1 iff n = 2^k: a(A000079(n)) = 1, a(A057716(n)) > 1.

Links

Programs

  • Haskell
    a225843 = floor . sum . map (recip . fromIntegral) . a070165_row

Formula

a(n) = floor(A225761(n)/A225784(n)).

A225878 Numbers n such that the products n*(sum of the reciprocals of the Collatz (3x+1) sequence beginning at n) are integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 80, 128, 160, 208, 256, 320, 416, 512, 640, 832, 1024, 1280, 1344, 1664, 2048, 2560, 2688, 3328, 4096, 5120, 5376, 6656, 8192, 10240, 10752, 13312, 16384, 20480, 21504, 21760, 26624, 27264, 32768, 40960, 43008, 43520, 53248, 54528
Offset: 1

Views

Author

Michel Lagneau, May 19 2013

Keywords

Comments

Numbers n such that A225784(n) divides n.
The powers of 2 are in the sequence, but there exists a subsequence of non-powers of 2: 80, 160, 208, 320, 416, 640, 832, 1280, 1344,... where the members are of the forms 5*2^p with p>=4, 13*2^p with p>=4, 21*2^p with p>=6, 213*2^p with p>=7, 341*2^p with p>=10,...

Examples

			208 is in the sequence because 208 *(1/208 + 1/104 + 1/52 + 1/26 + 1/13 + 1/40 + 1/20 + 1/10 + 1/5 + 1/16 + 1/8 + 1/4 +1/2 + 1/1) = 512 is an integer.
2688 is in the sequence because A225784(2688) = 896 divides 2688.

Crossrefs

Cf. A225784, A225761.

Programs

  • Mathematica
    collatz[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&];Select[Range[50000],IntegerQ[Total[#/collatz[#]]]&]
Showing 1-4 of 4 results.

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

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