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A375208 Modified Sisyphus function of order 5.

%I A375208 #24 Jan 05 2025 19:51:42
%S A375208 110000,101000,100100,100010,100001,110000,101000,100100,100010,
%T A375208 100001,211000,202000,201100,201010,201001,211000,202000,201100,
%U A375208 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
%N A375208 Modified Sisyphus function of order 5.
%C A375208 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.
%C A375208 If we start with n and repeatedly apply the map i -> a(i), we eventually get the cycle {613200, 622110}.
%H A375208 M. E. Coppenbarger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/56-2/coppenbarger03242018.pdf">Iterations of a modified Sisyphus function</a>, Fib. Q., 56 (No. 2, 2018), 130-141.
%e A375208 11 has two digits, both congruent to 1 modulo 5, so a(11) = 202000.
%e A375208 a(20) = 210100.
%e A375208 a(30) = 210010.
%e A375208 a(2527200000) = 1060400.
%p A375208 a:= n-> (l-> parse(cat(nops(l), seq(add(`if`(irem(i, 5)=k
%p A375208           , 1, 0), i=l), k=0..4))))(convert(n, base, 10)):
%p A375208 seq(a(n), n=0..44);  # _Alois P. Heinz_, Oct 23 2024
%o A375208 (Python) # based on _Michael S. Branicky_ in A350709
%o A375208 def a(n, order=5):
%o A375208     d, m = list(map(int, str(n))), [0]*order
%o A375208     for di in d: m[di%order] += 1
%o A375208     return int(str(len(d)) + "".join(map(str, m)))
%o A375208 print([a(n) for n in range(37)])
%o A375208 (Python)
%o A375208 from collections import Counter
%o A375208 def A375208(n):
%o A375208     s = str(n)
%o A375208     c = Counter(int(d)%5 for d in s)
%o A375208     return int(str(len(s))+''.join(str(c[i]) for i in range(5))) # _Chai Wah Wu_, Nov 26 2024
%Y A375208 Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A350709.
%K A375208 nonn,base,easy
%O A375208 0,1
%A A375208 _Matt Coppenbarger_, Oct 16 2024