A151962 -id:A151962 - cp's OEIS frontend

A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Joseph Myers, Aug 29 2009

All base-2 cycles are fixed points, so one less than A164886.

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A164884, A163205, A164886, A164887.

In other bases: A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

Cross-references edited by Joseph Myers, Sep 04 2009

A151963 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the Kaprekar map in A151949.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3, 2, 3, 7, 5, 6, 4, 4, 6, 5, 7, 3

Offset: 0

N. J. A. Sloane, Aug 19 2009

Equals A151962(n) + 1 iff n < 10001 (when a cycle of length greater than 1 occurs for the first time).

			13->18->63->27->45->9->0->0, so a(13)=6+1 = 7.

Joseph Myers and Robert G. Wilson v, Table of n, a(n) for n = 0..1000 .
Index entries for the Kaprekar map

In other bases: A164886 (base 2), A164996 (base 3), A165015 (base 4), A165035 (base 5), A165054 (base 6), A165074 (base 7), A165093 (base 8), A165113 (base 9). - Joseph Myers, Sep 05 2009

# Maple program from R. J. Mathar:
A151949 := proc(n)
local tup;
tup := sort(convert(n,base,10)) ;
add( (op(i,tup)-op(-i,tup)) *10^(i-1),i=1..nops(tup)) :
end:
A151963 := proc(n)
local tra,x ;
tra := [n] ;
x := n ;
while true do
x := A151949(x) ;
if x in tra then
RETURN(nops(tra)) ;
fi;
tra := [op(tra),x] :
od:
end:
seq(A151963(n),n=0..120) ;

f[n_] := Module[{idn = IntegerDigits@n, idns}, idns = Sort@ idn; FromDigits@ Reverse@ idns - FromDigits@ idns]; g[n_] := Length[ NestWhileList[ f, n, UnsameQ, All]] - 1; Table[g@n, {n, 0, 104}] (* Robert G. Wilson v, Aug 20 2009 *)

Typos corrected by Joseph Myers, Aug 20 2009

More terms from R. J. Mathar and Robert G. Wilson v, Aug 20 2009

A164995 Length of preperiodic part of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2187
Index entries for the Kaprekar map

Cf. A164993, A164996.

In other bases: A164885 (base 2), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A165014 Length of preperiodic part of trajectory of n under iteration of the base-4 Kaprekar map in A165012.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n = 0..1024
Index entries for the Kaprekar map

Cf. A165012, A165015.

In other bases: A164885 (base 2), A164995 (base 3), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A165034 Length of preperiodic part of trajectory of n under iteration of the base-5 Kaprekar map in A165032.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..3125
Index entries for the Kaprekar map

Cf. A165032, A165035.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165053 (base 6), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A165053 Length of preperiodic part of trajectory of n under iteration of the base-6 Kaprekar map in A165051.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 4, 3, 3, 4, 2, 1, 2, 4, 3, 3, 4, 2, 1, 2, 4, 3, 3, 4, 2, 1, 2, 4, 3, 3, 4, 2, 1, 2, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 1, 3, 3, 3, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 1, 2, 3, 3, 2, 2, 3, 1, 2, 3, 2, 1, 2, 3, 1, 1, 3, 2, 2, 3, 1, 2, 1, 3, 3, 3, 1, 3, 2, 1

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..1296
Index entries for the Kaprekar map

Cf. A165051, A165054.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165034 (base 5), A165073 (base 7), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A165073 Length of preperiodic part of trajectory of n under iteration of the base-7 Kaprekar map in A165071.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 4, 2, 4, 3, 2, 1, 2, 3, 4, 2, 4, 3, 2, 1, 2, 3, 4, 2, 4, 3, 2, 1, 2, 3, 4, 2, 4, 3, 2, 1, 2, 3, 4, 2, 4, 3, 2, 1, 2, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 1, 1, 2, 3, 2, 2, 3, 1, 1, 2, 1, 3, 3, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 3

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..2401
Index entries for the Kaprekar map

Cf. A165071, A165074.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165092 (base 8), A165112 (base 9), A151962 (base 10).

A165092 Length of preperiodic part of trajectory of n under iteration of the base-8 Kaprekar map in A165090.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 4, 1, 3, 3, 2, 4, 2, 1, 2, 4, 0, 3, 3, 1, 4, 2, 1, 2, 4, 1, 3, 3, 1, 4, 2, 1, 2, 4, 1, 3, 3, 1, 4, 2, 1, 2, 4, 2, 3, 3, 1, 4, 2, 1, 2, 4, 2, 3, 3, 1, 4, 2, 1, 2, 2, 4, 3, 1, 2, 3, 4, 2, 1, 2, 4, 3, 1, 2, 3, 4, 2, 2, 4, 3, 1, 2, 3, 3, 4, 4, 4, 3, 1, 2, 3, 1, 3, 3, 3, 3, 1, 2, 3, 2

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..4096
Index entries for the Kaprekar map

Cf. A165090, A165093.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165112 (base 9), A151962 (base 10).

A165112 Length of preperiodic part of trajectory of n under iteration of the base-9 Kaprekar map in A165110.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 0, 3, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 0, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 4, 3, 1, 1, 2, 3, 4, 2, 1, 2, 4, 3, 1, 1, 2, 3, 4, 2, 2, 4, 3, 1

Offset: 0

Joseph Myers, Sep 04 2009

Joseph Myers, Table of n, a(n) for n=0..6561
Index entries for the Kaprekar map

Cf. A165110, A165113.

In other bases: A164885 (base 2), A164995 (base 3), A165014 (base 4), A165034 (base 5), A165053 (base 6), A165073 (base 7), A165092 (base 8), A151962 (base 10).

A343383 Length of the preperiodic part of 'Roll and Subtract' trajectory of n.

Original entry on oeis.org

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

Offset: 0

Jonathon Priestley, Apr 12 2021

'Roll and Subtract' is defined by x -> |x - roll(x)|, where roll(x) takes the first digit of a number and moves it to the back (rolls it around to the back).

Differs from A151962 first at n=101. - R. J. Mathar, May 07 2021

			a(119) = 4 since |119 - 191| = 72 -> |72 - 27| = 45 -> |45 - 54| = 9 -> |9 - 9| = 0. The value a(0) maps to 0, so the sequence ends there after 4 values have been traversed.
a(12737) = 1 since |12737 - 27371| = 14634 -> |14634 - 46341| = 31707 -> |31707 - 17073| = 14634. Since 14634 is already in the sequence, the sequence ends there.

Jonathon Priestley, Table of n, a(n) for n = 0..9999

Cf. A072137 (reverse and subtract).

Array[Function[w, LengthWhile[w, # != Last[w] &]]@ NestWhileList[Abs[# - FromDigits@ RotateLeft@ IntegerDigits[#]] &, #, Unequal, All] &, 105, 0] (* Michael De Vlieger, Apr 13 2021 *)

def roll(n):
    """ Moves first digit to the back """
    s = str(n)
    return int(s[1:] + s[0])
def backtrack(past, length, offset, dct):
    """ Goes through every value passed and adds it and it's length to the dictionary """
    if length == 0:
        for elem in past:
            dct[elem] = 0
    i = 0
    while length > 0:
        n = past[i]
        dct[n] = length + offset
        i += 1
        length -= 1
    return dct
def a(n, dct):
    past = []
    length = 0
    while (n not in dct):
        past.append(n)
        length += 1
        n = abs(n - roll(n))
        if n in past: # For duplicates
            length = past.index(n)
            dct = backtrack(past, length, 0, dct)
            return dct, length
    offset = dct[n]
    dct = backtrack(past, length, offset, dct)
    length += offset
    return dct, length
dct = {}
sequence = []
i = 1
while i < 1000:
    out = a(i, dct)
    dct = out[0]
    sequence.append(out[1])
    i += 1

cp's OEIS Frontend

A164885 Length of preperiodic part of trajectory of n under iteration of the base-2 Kaprekar map in A164884.

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A151963 (Length of preperiodic part) + (length of cycle) of trajectory of n under iteration of the Kaprekar map in A151949.

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Mathematica

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A164995 Length of preperiodic part of trajectory of n under iteration of the base-3 Kaprekar map in A164993.

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A165014 Length of preperiodic part of trajectory of n under iteration of the base-4 Kaprekar map in A165012.

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A165034 Length of preperiodic part of trajectory of n under iteration of the base-5 Kaprekar map in A165032.

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A165053 Length of preperiodic part of trajectory of n under iteration of the base-6 Kaprekar map in A165051.

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A165073 Length of preperiodic part of trajectory of n under iteration of the base-7 Kaprekar map in A165071.

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A165092 Length of preperiodic part of trajectory of n under iteration of the base-8 Kaprekar map in A165090.

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A165112 Length of preperiodic part of trajectory of n under iteration of the base-9 Kaprekar map in A165110.

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A343383 Length of the preperiodic part of 'Roll and Subtract' trajectory of n.

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Mathematica

Python