A375208 - OEIS

A375208

Modified Sisyphus function of order 5: a(n) is the concatenation of the number of digits in n with the number of digits of n congruent to k modulo 5 for each k from 0 to 4 in turn.

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

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OFFSET

0,1

COMMENTS

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

LINKS

James C. McMahon, Table of n, a(n) for n = 0..10000

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

EXAMPLE

11 has two digits, both congruent to 1 modulo 5, so a(11) = 202000.

a(20) = 210100.

a(30) = 210010.

a(2527200000) = 1060400.

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

MATHEMATICA

fc[n_, m_]:=Count[Mod[IntegerDigits[n], 5], m]; a[n_]:=Module[{s={Length[IntegerDigits[n]]}}, Do[AppendTo[s, fc[n, k]], {k, 0, 4}]; FromDigits[s]]; Array[a, 45, 0] (* James C. McMahon, Sep 07 2025 *)

PROG

(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)