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 1-5 of 5 results.

A087227 Number of distinct prime factors of A087226(n), the LCM of terms in trajectory of 3x+1 (function) initiated at n.

Original entry on oeis.org

0, 1, 3, 1, 2, 3, 6, 1, 7, 2, 5, 3, 3, 6, 6, 1, 4, 7, 7, 2, 3, 5, 5, 3, 7, 3, 37, 6, 6, 6, 35, 1, 8, 4, 4, 7, 7, 7, 12, 2, 36, 3, 8, 5, 5, 5, 35, 3, 7, 7, 8, 3, 3, 37, 35, 6, 10, 6, 11, 6, 6, 35, 36, 1, 7, 8, 9, 4, 5, 4, 34, 7, 36, 7, 5, 7, 7, 12, 12, 2, 7, 36, 35, 3, 3, 8, 11, 5, 10, 5, 32, 5, 6, 35, 35
Offset: 1

Views

Author

Labos Elemer, Aug 28 2003

Keywords

Comments

This sequence differs from A078719: a(n) <= (number of odd terms in list).

Examples

			n=25: A087226(25) = 535792400 with factor-set={2,5,11,13,17,19,29} so a(25)=7, while number of odd terms in the trajectory is A078219(25)=8.

Crossrefs

Cf. A006370, A087226, A078719.

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

A178168 Product of the numbers in the Collatz (3x+1) trajectory of n, including n.

Original entry on oeis.org

1, 2, 153600, 8, 5120, 921600, 704889816350720000, 64, 2486851272085340160000, 51200, 4577206599680000, 11059200, 532480000, 9868457428910080000, 114523513552896000000000, 1024, 12238520320000, 44763322897536122880000
Offset: 1

Views

Author

T. D. Noe, May 21 2010

Keywords

Comments

Row n of A070165 has the Collatz trajectory of n. It appears that all products are unique. This has been verified for all n < 10^6 and for the 12332052 values of n for which a(n) < 2^1081.

Examples

			The Collatz iteration starting with 3 is 3, 10, 5, 16, 8, 4, 2, 1. The product of these numbers is 153600.

Links

Crossrefs

Cf. A087226 (LCM of the trajectory), A178169, A178170

Programs

  • Mathematica
    Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Table[Times@@Collatz[n], {n, 100}]

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

A087228 a(n) is the smallest number k such that the LCM of the terms of the Collatz trajectory of k has n distinct prime factors.

Original entry on oeis.org

2, 5, 3, 17, 11, 7, 9, 33, 67, 57, 59, 39, 105, 185, 191, 123, 225, 219, 239, 159, 319, 283, 251, 167, 335, 111, 297, 175, 233, 155, 103, 91, 107, 71, 31, 41, 27, 193, 129, 231, 171, 463, 327, 411, 859, 731, 487, 649, 639, 1153, 1563, 1607, 1071, 1215, 1307, 871, 1161
Offset: 1

Views

Author

Labos Elemer, Aug 28 2003

Keywords

Examples

			a(10)=57 because 57 is the smallest number such that the LCM of the terms in its Collatz trajectory has 10 different prime factors: A082226(57) = 864203580240 = 2^4*3*5*7*11*13*17*19*37*43.

Crossrefs

Cf. A006370, A087226, A086227, A078719, A001221.

Programs

  • 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] ef[x_] := Length[FactorInteger[Apply[LCM, fpl[x]]]] t=Table[0, {256}]; Do[s=ef[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 1000}]; t

Formula

a(n) = Min{k; A087227(k)=n}, where A087227(k) = A001221(A087226(k)); A087226(k) = lcm(terms in Collatz trajectory of k).

Extensions

Edited by Jon E. Schoenfield, Jul 09 2018
Showing 1-5 of 5 results.

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