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

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

0,11

Comments

'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

Links

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

Example

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.

Mathematica

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

Prog

(Python)
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

Crossrefs

Cf. A072137 (reverse and subtract).

Sequence in context: A174959 A126093 A065279 * A151962 A072137 A061569

Adjacent sequences: A343380 A343381 A343382 * A343384 A343385 A343386

Keyword

nonn,base

Author

Jonathon Priestley, Apr 12 2021

Status

approved