A099009 -id:A099009 - cp's OEIS frontend

A164733 Number of n-digit fixed points under the Kaprekar map A151949.

1, 0, 1, 1, 0, 2, 0, 2, 2, 3, 1, 5, 1, 6, 2, 8, 2, 12, 3, 14, 5, 17, 7, 21, 8, 25, 12, 30, 14, 36, 17, 43, 21, 49, 25, 58, 31, 66, 36, 75, 43, 85, 49, 96, 58, 109, 66, 121, 75, 136, 86, 150, 96, 167, 109, 184, 121, 202, 136, 222, 150, 242, 167, 265, 185, 287, 202, 313, 222, 338

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..70
Index entries for the Kaprekar map

Cf. A151949, A099009, A164731, A164732, A164734, A164735, A164736.
Bisections: A309223, A309224.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A165027 (base 4), A008617 (base 5), A165066 (base 6), A008722 (base 7, conjecturally), A165105 (base 8), A165125 (base 9). [From Joseph Myers, Sep 05 2009]

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-2) + a(n-6) - a(n-8) + a(n-9) - a(n-11) + a(n-14) - a(n-15) - a(n-16) + a(n-17) - a(n-20) + a(n-22) - a(n-23) + a(n-25) + a(n-29) - a(n-31) for n > 33.
G.f.: x*(-x^32 + x^31 - x^29 + x^28 - x^27 + x^26 - x^24 + 2*x^23 - x^22 + x^21 + x^20 + 2*x^18 - x^17 + x^16 + 2*x^15 - 3*x^14 + 2*x^13 - x^12 + x^11 - x^9 + 2*x^8 - x^6 + x^5 - x^4 + x^3 + 1)/(x^31 - x^29 - x^25 + x^23 - x^22 + x^20 - x^17 + x^16 + x^15 - x^14 + x^11 - x^9 + x^8 - x^6 - x^2 + 1). (End)

A055162 The number n has digits in ascending order and n=a-b where a has the digits of n in descending order and b has the digits of n in another order (a and b perhaps with extra zeros), ordered by a.

Original entry on oeis.org

459, 1467, 445599, 134667, 13346667, 1245789, 123456789, 444555999, 1333466667, 123456789, 12457899, 12334566789, 133334666667, 12334566789, 1234567899, 124578999, 444455559999, 1233345666789

Offset: 1

Henry Bottomley, Apr 27 2000

Each term of this sequence A055162(n) corresponds to A099009(n+1), with its digits being reordered in the ascending manner. - Alexander R. Povolotsky, Apr 26 2012

All terms of this sequence are divisible by nine, yielding 51, 163, 49511, 14963, 1482963,.... - Alexander R. Povolotsky, Apr 29 2012

			459 = 954 - 495.

Denis Borris, Ken Duisenberg's Puzzle of the Week, April 20, 2000

Source

Cf. A055159, A099009.

a(n) = A055161(n) - A055160(n).

A164723 Numbers belonging to cycles of length 2 under the Kaprekar map A151949.

Original entry on oeis.org

53955, 59994, 8733209876622, 9665429654331, 873332098766622, 966543296654331, 8764421997755322, 8765431997654322, 87333320987666622, 96654332966654331, 8733333209876666622, 9665433329666654331

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..1146
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164725, A164727, A164729, A164724.

A164724 Least element of each cycle of length 2 under the Kaprekar map A151949.

Original entry on oeis.org

53955, 8733209876622, 873332098766622, 8764421997755322, 87333320987666622, 8733333209876666622, 873333332098766666622, 87333333320987666666622, 87764442219997775553222, 8733333333209876666666622

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..573
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A164718, A164720, A164726, A164728, A164730, A164723.

A164725 Numbers belonging to cycles of length 3 under the Kaprekar map A151949.

Original entry on oeis.org

64308654, 83208762, 86526432, 6431088654, 6433086654, 6543086544, 8321088762, 8332087662, 8653266432, 8655264432, 8732087622, 8765264322, 9751088421, 9755084421, 9775084221, 643110888654, 643310886654, 643330866654

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..27849
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164727, A164729, A164726.

A164726 Least element of each cycle of length 3 under the Kaprekar map A151949.

Original entry on oeis.org

64308654, 6431088654, 6433086654, 6543086544, 9751088421, 643110888654, 643310886654, 643330866654, 654310886544, 654330866544, 655430865444, 975110888421, 975310886421, 975510884421, 997510884201, 64311108888654

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..9283
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A164718, A164720, A164724, A164728, A164730, A164725.

A164727 Numbers belonging to cycles of length 5 under the Kaprekar map A151949.

Original entry on oeis.org

86420987532, 86541975432, 87641975322, 88431976512, 96641975331, 8643209876532, 8654209875432, 8654319765432, 8764209875322, 8764319765322, 8765419754322, 8843209876512, 8843319766512, 8854319765412, 8874319765212

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..14510
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164725, A164729, A164728.

A164728 Least element of each cycle of length 5 under the Kaprekar map A151949.

Original entry on oeis.org

86420987532, 8643209876532, 8654209875432, 8764209875322, 864332098766532, 865432098765432, 876432098765322, 876542098754322, 885432098765412, 86433320987666532, 86543320987665432, 87643320987665322

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..2902
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A164718, A164720, A164724, A164726, A164730, A164727.

A055160 Numbers n with the property that n=a-b where a has the digits of n in descending order and b has the digits of n in ascending order (perhaps with leading zeros), ordered by a.

Original entry on oeis.org

495, 6174, 549945, 631764, 63317664, 97508421, 864197532, 554999445, 6333176664, 9753086421, 9975084201, 86431976532, 633331766664, 975330866421, 997530864201, 999750842001, 555499994445, 8643319766532

Offset: 1

Henry Bottomley, Apr 27 2000

This sequence appears to differ from A099009 at the positions of two terms in it: 554999445 and 555499994445. - Alexander R. Povolotsky, May 01 2012

			495 = 954-459.

Denis Borris, Ken Duisenberg's Puzzle of the Week, April 20, 2000

Denis Borris, Ken Duisenberg's Puzzle of the Week, April 20, 2000
Yutaka Nishiyama, The weirdness of number 6174, International Journal of Pure and Applied Mathematicsm, Volume 80 No. 3 2012, 363-373. - From _N. J. A. Sloane_, Oct 29 2014

Cf. A055157, A099009.

a(n) = A055161(n) - A055162(n).

A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

Original entry on oeis.org

420876, 642654, 750843, 840852, 851742, 860832, 862632, 43208766, 64326654, 75308643, 84308652, 85317642, 86308632, 86326632, 4332087666, 6433266654, 7533086643, 8433086652, 8533176642, 8633086632, 8633266632, 433320876666

Offset: 1

Joseph Myers, Aug 23 2009

Joseph Myers, Table of n, a(n) for n=1..476
Index entries for the Kaprekar map

Cf. A151949, A151959, A099009, A099010, A164716, A164723, A164725, A164727, A164730.

cp's OEIS Frontend

A164733 Number of n-digit fixed points under the Kaprekar map A151949.

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A055162 The number n has digits in ascending order and n=a-b where a has the digits of n in descending order and b has the digits of n in another order (a and b perhaps with extra zeros), ordered by a.

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A164723 Numbers belonging to cycles of length 2 under the Kaprekar map A151949.

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A164724 Least element of each cycle of length 2 under the Kaprekar map A151949.

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A164725 Numbers belonging to cycles of length 3 under the Kaprekar map A151949.

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A164726 Least element of each cycle of length 3 under the Kaprekar map A151949.

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A164727 Numbers belonging to cycles of length 5 under the Kaprekar map A151949.

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A164728 Least element of each cycle of length 5 under the Kaprekar map A151949.

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A055160 Numbers n with the property that n=a-b where a has the digits of n in descending order and b has the digits of n in ascending order (perhaps with leading zeros), ordered by a.

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A164729 Numbers belonging to cycles of length 7 under the Kaprekar map A151949.

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