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 1-3 of 3 results.

A225784 Denominators of the sum of the reciprocals of the Collatz (3x+1) sequence beginning at n.

Original entry on oeis.org

1, 2, 240, 4, 80, 80, 272272, 8, 350064, 80, 38896, 240, 208, 272272, 4095840, 16, 3536, 116688, 21431696, 80, 1344, 38896, 1365280, 80, 535792400, 208, 44841486948146266934850832405421294927083491752830032389039800908293040266400, 38896, 1127984, 1365280
Offset: 1

Views

Author

Nico Brown, May 15 2013

Keywords

Comments

If the sum of the reciprocals of a Collatz sequence is bounded, there are no Collatz cycles other than 4,2,1,4,2,1,...
a(n) = denominator of Sum_{k = 1..A006577(n)} 1/A070165(n,k). - 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 denominator is 350064.

Links

Crossrefs

Cf. A225761 (numerators), A087226.
Cf. A225843.

Programs

  • Haskell
    import Data.Ratio (denominator)
a225784 = denominator . sum . map (recip . fromIntegral) . a070165_row
-- Reinhard Zumkeller, May 16 2013
  • Mathematica
    Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Denominator[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 1-3 of 3 results.

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

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