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.

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A165041 Consider the base-5 Kaprekar map n->K(n) defined in A165032. Sequence gives least elements of each cycle, including fixed points.

Original entry on oeis.org

0, 8, 48, 392, 1992, 7488, 53712, 249992, 1831056, 6249992, 45781056, 48217776, 170312312, 1144531056, 1205467776, 1217651376, 4514058432, 4576557032, 22460937432, 28613281056, 28671874056, 30136717776, 30441401376
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Comments

Initial terms in base 5: 0, 13, 143, 3032, 30432, 214423, 3204322, 30444432, 432043211, 3044444432.

Links

Crossrefs

Union of A165036 and A165043. Cf. A165032, A165042, A165037, A165039, A165049, A165045.
In other bases: A163205 (base 2), A165002 (base 3), A165021 (base 4), A165060 (base 6), A165080 (base 7), A165099 (base 8), A165119 (base 9), A164718 (base 10).

A165043 Consider the base-5 Kaprekar map n->K(n) defined in A165032. Sequence gives least elements of each cycle of length > 1.

Original entry on oeis.org

48, 1992, 7488, 53712, 249992, 6249992, 45781056, 170312312, 1144531056, 1205467776, 4514058432, 4576557032, 22460937432, 28613281056, 28671874056, 30136717776, 30441401376, 106445312312, 112304687432, 715332031056
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Comments

Initial terms in base 5: 143, 30432, 214423, 3204322, 30444432, 3044444432, 43204443211, 322044443222, 4320444443211, 4432044432101.

Links

Crossrefs

Cf. A165032, A165044, A165036, A165037, A165041, A165039, A165049.
In other bases: Empty (base 2), A165004 (base 3), A165023 (base 4), A165062 (base 6), A165082 (base 7), A165101 (base 8), A165121 (base 9), A164720 (base 10).

A008617 Expansion of 1/((1-x^2)(1-x^7)).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of (n+9)-digit fixed points under the base-5 Kaprekar map A165032 (see A165036 for the list of fixed points). - Joseph Myers, Sep 04 2009
It appears that this is the number of partitions of 4*n that are 8-term arithmetic progressions. Further, it seems that a(n)=[n/2]-[3n/7]. - John W. Layman, Feb 21 2012

References

  • D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Links

Programs

  • Mathematica
    CoefficientList[Series[1 / ((1 - x^2) (1 - x^7)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 22 2013 *)
LinearRecurrence[{0,1,0,0,0,0,1,0,-1},{1,0,1,0,1,0,1,1,1},80] (* Harvey P. Dale, May 18 2018 *)

Formula

a(n) = floor((2*n+21+7*(-1)^n)/28). - Tani Akinari, May 19 2014

Extensions

Typo in name fixed by Vincenzo Librandi, Jun 22 2013

A165047 Consider the base-5 Kaprekar map x->K(x) described in A165032. Sequence gives the smallest number that belongs to a cycle of length n under repeated iteration of this map, or -1 if there is no cycle of length n.

Original entry on oeis.org

0, 48, 45781056, 1992, 7488, 249992, 26648194761946797370910644531056, 170312312, 447082519531056, 953674316406249992, 43487548828124832, 68219378590583801269531056
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Comments

Known values (to 100 base-5 digits):
a(1) = 0 (base 10) = 0 (base 5)
a(2) = 48 (base 10) = 143 (base 5)
a(3) = 45781056 (base 10) = 43204443211 (base 5)
a(4) = 1992 (base 10) = 30432 (base 5)
a(5) = 7488 (base 10) = 214423 (base 5)
a(6) = 249992 (base 10) = 30444432 (base 5)
a(7) = 26648194761946797370910644531056 (base 10) = 432044444444444444444444444444444444444443211 (base 5)
a(8) = 170312312 (base 10) = 322044443222 (base 5)
a(9) = 447082519531056 (base 10) = 432044444444444443211 (base 5)
a(10) = 953674316406249992 (base 10) = 30444444444444444444444432 (base 5)
a(11) = 43487548828124832 (base 10) = 331044444444444444443312 (base 5)
a(12) = 68219378590583801269531056 (base 10) = 4320444444444444444444444444444443211 (base 5)
a(13) = 388774887899923005107893914100714027881532907485961914056 (base 10) = 432222222222222222222222044444444444444444444444444444443222222222222222222222211 (base 5)
a(14) = 4366040229797363281056 (base 10) = 4320444444444444444444444443211 (base 5)
a(15) = 15550995515996920287582474884401599410921335220336914056 (base 10) = 4322222222222222222222222222222222044444444432222222222222222222222222222222211 (base 5)
a(18) = 1705484464764595031738281056 (base 10) = 432044444444444444444444444444444443211 (base 5)
a(20) = 6505906924303417326882481575012207031056 (base 10) = 432044444444444444444444444444444444444444444444444443211 (base 5)
a(21) = 416378043155418708920478820800781056 (base 10) = 432044444444444444444444444444444444444444444443211 (base 5)
a(22) = 39708904567281599895522958831861615180969238281056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(23) = 16655121726216748356819152832031056 (base 10) = 4320444444444444444444444444444444444444444443211 (base 5)
a(24) = 1479271969470441337508073405618280737883196707116439938545227050781056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(26) = 260236276972136693075299263000488281056 (base 10) = 4320444444444444444444444444444444444444444444444443211 (base 5)
a(27) = 9694556779121484349492909871059964643791317939758300781056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(28) = 151477449673773192960826716735311947559239342808723449707031056 (base 10) = 43204444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(29) = 4066191827689635829301550984382629394531056 (base 10) = 4320444444444444444444444444444444444444444444444444444443211 (base 5)
a(30) = 101654795692240895732538774609565734863281056 (base 10) = 432044444444444444444444444444444444444444444444444444444443211 (base 5)
a(33) = 1588356182691263995820918353274464607238769531056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(35) = 992722614182039997388073970796540379524230957031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(36) = 59170878778817653500322936224731229515327868284657597541809082031056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(39) = 387782271164859373979716394842398585751652717590332031056 (base 10) = 432044444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)
a(41) = 242363919478037108737322746776499116094782948493957519531056 (base 10) = 4320444444444444444444444444444444444444444444444444444444444444444444444444444443211 (base 5)

Links

Crossrefs

Cf. A165032, A165036, A165037, A165039, A165041, A165043.
In other bases: A153881 (base 2), A165008 (base 3), A165028 (base 4), A165067 (base 6), A165086 (base 7), A165106 (base 8), A165126 (base 9), A151959 (base 10).

A319798 Smallest fixed points (>0) of the base-n Kaprekar map.

Original entry on oeis.org

9, 184, 30, 8, 105, 1922263344, 21, 41520, 495, 40, 858, 3488424, 65, 30996, 2040, 96, 2907, 264925230120, 133, 2787400, 5313, 176, 6900, 237360, 225, 9742824, 10962, 280, 13485, 763713003420, 341, 26485184, 19635, 408, 23310, 107599353444576, 481, 60920080, 31980
Offset: 2

Views

Author

Seiichi Manyama, Sep 28 2018

Keywords

Comments

Conjecture: If n = 3*k - 1 (>2), a(n) = A000567(k). For example, a(29) = 10 * (3*10 - 2) = 280.

Links

Crossrefs

Cf. A000567, A099009, A163205, A164997, A165016, A165036, A165055, A165075, A165094, A165114, A319839.

Extensions

a(19) and a(31)-a(40) from Giovanni Resta, Oct 02 2018
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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