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A383931 Minimal nonnegative integer which reaches a cycle after exactly n iterations of the modified Sisyphus function of order 5 (A375208).

Original entry on oeis.org

613200, 100123, 100012, 10, 1023, 100, 0, 10234, 10000123, 10000000000002

Offset: 0

Matt Coppenbarger, May 15 2025

The sole cycle is C = {613200,622110} and iteration stops on reaching either element of this cycle.

The next term is 145 digits a(10) = decimal 1 0^(101) 1^(10) 2^(11) 3^(11) 4^(11) where ^ denotes repetition of a digit.

			For n=6, a(6) = 0 takes 6 iterations to reach C: 0 -> 110000 -> 642000 -> 631101 -> 614010 -> 632001 -> 6221100.

J. Schram, The Sisyphus string, J. Recreational Math., 19 (1987), 43-44.

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Cf. A375208 (Sisyphus 5 function).

Cf. A308002, A352752.

A375208 Modified Sisyphus function of order 5.

Original entry on oeis.org

110000, 101000, 100100, 100010, 100001, 110000, 101000, 100100, 100010, 100001, 211000, 202000, 201100, 201010, 201001, 211000, 202000, 201100, 201010, 201001, 210100, 201100, 200200, 200110, 200101, 210100, 201100, 200200, 200110, 200101, 210010, 201010, 200110, 200020, 200011, 210010, 201010, 200110, 200020, 200011, 210001, 201001, 200101, 200011, 200002

Offset: 0

Matt Coppenbarger, Oct 16 2024

a(n) is the concatenation of the number of digits in n with number of digits of n congruent to k modulo 5 for each k from 0 to 4 in turn. See Example.

If we start with n and repeatedly apply the map i -> a(i), we eventually get the cycle {613200, 622110}.

			11 has two digits, both congruent to 1 modulo 5, so a(11) = 202000.
a(20) = 210100.
a(30) = 210010.
a(2527200000) = 1060400.

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A350709.

a:= n-> (l-> parse(cat(nops(l), seq(add(`if`(irem(i, 5)=k
          , 1, 0), i=l), k=0..4))))(convert(n, base, 10)):
seq(a(n), n=0..44);  # Alois P. Heinz, Oct 23 2024

# based on Michael S. Branicky in A350709
def a(n, order=5):
    d, m = list(map(int, str(n))), [0]*order
    for di in d: m[di%order] += 1
    return int(str(len(d)) + "".join(map(str, m)))
print([a(n) for n in range(37)])

from collections import Counter
def A375208(n):
    s = str(n)
    c = Counter(int(d)%5 for d in s)
    return int(str(len(s))+''.join(str(c[i]) for i in range(5))) # Chai Wah Wu, Nov 26 2024

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User: Matt Coppenbarger

A383931 Minimal nonnegative integer which reaches a cycle after exactly n iterations of the modified Sisyphus function of order 5 (A375208).

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