A165071 -id:A165071 - cp's OEIS frontend

A165088 Smallest member of cycle corresponding to n-th term of A165087.

0, 144, 1068, 9936, 55500, 640992, 3562968, 31412208, 228238488, 227429400, 220709400, 1922263344, 11432420652, 11150046252, 75796404672, 94197649008, 96503566608, 546394287000, 419850417612, 4615731883344, 3939440152944

Offset: 1

Joseph Myers, Sep 04 2009

Initial terms in base 7: 0, 264, 3054, 40653, 320544, 5306532, 42166443, 530666532, 5440665222, 5431055322.

Joseph Myers, Table of n, a(n) for n=1..13070
Index entries for the Kaprekar map

Cf. A165071, A165076, A165087, A165089.

In other bases: A163205 (base 2), A165010 (base 3), A165030 (base 4), A165049 (base 5), A165069 (base 6), A165108 (base 8), A165128 (base 9), A151965 (base 10).

A165089 Length of cycle corresponding to n-th term of A165087.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 1, 5, 2, 6, 1, 1, 2, 9, 1, 2, 1, 1, 12, 2, 2, 1, 1, 1, 1, 2, 2, 1, 4, 2, 1, 1, 1, 1, 5, 7, 1, 2, 2, 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 5, 2, 1, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 5, 5, 2, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 24, 1, 1, 2, 1, 1

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..13070
Index entries for the Kaprekar map

Cf. A165071, A165076, A165087, A165088.

In other bases: A000012 (base 2), A165011 (base 3), A165031 (base 4), A165050 (base 5), A165070 (base 6), A165109 (base 8), A165129 (base 9), A151966 (base 10).

A165077 Length of cycle mentioned in A165076.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 2, 2, 2, 2, 2, 1, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 1, 6, 1, 9, 9, 9, 2, 2, 9, 9, 9, 9, 9, 9, 2, 1, 2, 1, 1, 2, 2, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..36122
Index entries for the Kaprekar map

Cf. A165071, A165076, A165079, A165081, A165083, A165089.

In other bases: A000012 (base 2), A164999 (base 3), A165018 (base 4), A165038 (base 5), A165057 (base 6), A165096 (base 8), A165116 (base 9), A164717 (base 10).

A165079 Length of cycle mentioned in A165078.

Original entry on oeis.org

2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 9, 9, 9, 2, 2, 9, 9, 9, 9, 9, 9, 2, 2, 2, 2, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 2, 2, 2, 2, 4, 4, 4, 2, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..32041
Index entries for the Kaprekar map

Cf. A165071, A165078, A165077, A165081, A165083, A165089.

In other bases: Empty (base 2), A165001 (base 3), A165020 (base 4), A165040 (base 5), A165059 (base 6), A165098 (base 8), A165118 (base 9), A164715 (base 10).

A165081 Length of cycle mentioned in A165080.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 1, 2, 5, 6, 1, 1, 9, 2, 2, 1, 1, 1, 2, 12, 1, 2, 1, 1, 1, 2, 4, 1, 2, 1, 2, 1, 1, 1, 2, 5, 7, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 5, 1, 2, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 5, 5, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 1, 1, 1, 2, 24, 1, 1, 1, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..13070
Anthony Kay and Katrina Downes-Ward, Fixed Points and Cycles of the Kaprekar Transformation: 1. Odd Bases, Journal of Integer Sequences, Vol. 25 (2022), Article 22.6.7.
Index entries for the Kaprekar map

Cf. A165071, A165080, A165077, A165079, A165083, A165089.

In other bases: A000012 (base 2), A165003 (base 3), A165022 (base 4), A165042 (base 5), A165061 (base 6), A165100 (base 8), A165120 (base 9), A164719 (base 10).

A165083 Length of cycle mentioned in A165082.

Original entry on oeis.org

2, 3, 5, 6, 6, 6, 11, 2, 2, 2, 2, 5, 6, 9, 2, 2, 2, 12, 2, 2, 4, 2, 2, 2, 5, 7, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 5, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 24, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 7, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 25, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2

Offset: 1

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=1..8989
Index entries for the Kaprekar map

Cf. A165071, A165082, A165077, A165079, A165081, A165089.

In other bases: Empty (base 2), A165005 (base 3), A165024 (base 4), A165044 (base 5), A165063 (base 6), A165102 (base 8), A165122 (base 9), A164721 (base 10).

A008722 Molien series for 3-dimensional group [2,9] = *229.

Original entry on oeis.org

1, 0, 2, 0, 3, 0, 4, 0, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 11, 21, 13, 23, 15, 25, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 30, 45, 33, 48, 36, 51, 39, 54, 42, 58, 45, 62, 48, 66, 51, 70, 54, 74, 58, 78, 62, 82, 66, 86, 70, 90, 74, 95, 78, 100, 82

Offset: 0

N. J. A. Sloane

It appears that a(n) is the number of (n+11)-digit fixed points under the base-7 Kaprekar map A165071 (see A165075 for the list of fixed points). - Joseph Myers, Sep 04 2009

a(n) is the number of partitions of n into parts 2 and 9 where there are two kinds of parts 2. - Hoang Xuan Thanh, Jun 20 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 225
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,0,1,0,-2,0,1).

a:=[1,0,2,0,3,0,4,0,5,1,6,2,7];; for n in [14..80] do a[n]:= 2*a[n-2] -a[n-4]+a[n-9]-2*a[n-11]+a[n-13]; od; a; # G. C. Greubel, Sep 09 2019

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019

1/((1-x^2)^2*(1-x^9)); seq(coeff(series(%, x, n+1), x, n), n = 0..80); # modified by G. C. Greubel, Sep 09 2019

LinearRecurrence[{0,2,0,-1,0,0,0,0,1,0,-2,0,1}, {1,0,2,0,3,0,4,0,5,1,6, 2,7}, 80] (* Ray Chandler, Jul 15 2015 *)

my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^9))) \\ G. C. Greubel, Sep 09 2019

def A008722_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)^2*(1-x^9)) ).list()
A008722_list(80) # G. C. Greubel, Sep 09 2019

G.f.: 1/((1-x^2)^2*(1-x^9)).
a(n) = 2*a(n-2) - a(n-4) + a(n-9) - 2*a(n-11) + a(n-13). - R. J. Mathar, Dec 18 2014
a(n) = floor((n^2 + n*(13+9*(-1)^n) + 62*(-1)^n + 75)/72) - [(n mod 9)=7], where [] is Iverson bracket. - Hoang Xuan Thanh, Jun 20 2025

A319655 Write n in 7-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 9, 16, 17, 18, 19, 20, 3, 10, 17, 24, 25, 26, 27, 4, 11, 18, 25, 32, 33, 34, 5, 12, 19, 26, 33, 40, 41, 6, 13, 20, 27, 34, 41, 48, 1, 8, 9, 10, 11, 12, 13, 8, 57, 58, 59, 60, 61, 62, 9, 58, 65, 66, 67, 68, 69, 10, 59, 66, 73

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..2400

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), this sequence (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165071, A319724.

a:= n-> (l-> add(l[-i]*7^(i-1), i=1..nops(l)))(sort(convert(n, base, 7))):
seq(a(n), n=0..73);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 7]], 7], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 7)), 7); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(7, 100)

A319724 Write n in 7-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 22, 29, 36, 43, 14, 15, 16, 23, 30, 37, 44, 21, 22, 23, 24, 31, 38, 45, 28, 29, 30, 31, 32, 39, 46, 35, 36, 37, 38, 39, 40, 47, 42, 43, 44, 45, 46, 47, 48, 49, 56, 105, 154, 203, 252, 301, 56, 57, 106, 155, 204, 253, 302, 105, 106, 113, 162

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..2400

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), A319722 (b=5), A319723 (b=6), this sequence (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165071, A319655.

Table[FromDigits[ReverseSort[IntegerDigits[n, 7]], 7], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 7), , 4), 7); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(7, 100)

n <= a(n) < 7n. - Charles R Greathouse IV, Aug 07 2024

cp's OEIS Frontend

A165088 Smallest member of cycle corresponding to n-th term of A165087.

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A165089 Length of cycle corresponding to n-th term of A165087.

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A165077 Length of cycle mentioned in A165076.

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A165079 Length of cycle mentioned in A165078.

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A165081 Length of cycle mentioned in A165080.

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A165083 Length of cycle mentioned in A165082.

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A008722 Molien series for 3-dimensional group [2,9] = *229.

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A319655 Write n in 7-ary, sort digits into increasing order.

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Maple

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PARI

Ruby

A319724 Write n in 7-ary, sort digits into decreasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Ruby

Formula