A375208 Modified Sisyphus function of order 5.

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

Author

Matt Coppenbarger, Oct 16 2024

Keywords

• nonn
• base
• easy

Comments

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}.

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

Python

# 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)])

Python

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