A350709 Modified Sisyphus function of order 3: a(n) is the concatenation of the number of digits in n with the number of digits of n congruent to k modulo 3 for each k from 0 to 2 in turn.

1100, 1010, 1001, 1100, 1010, 1001, 1100, 1010, 1001, 1100, 2110, 2020, 2011, 2110, 2020, 2011, 2110, 2020, 2011, 2110, 2101, 2011, 2002, 2101, 2011, 2002, 2101, 2011, 2002, 2101, 2200, 2110, 2101, 2200, 2110, 2101, 2200

Offset

0,1

Comments

If we start with n and repeatedly apply the map i -> a(i), we eventually get the cycle {4031, 4112, 4220}

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 3, so a(11) = 2020.
a(20) = 2101.
a(30) = 2200.
a(1111123567) = 10262.

Maple

nd:= n -> 1 + ilog10(n):
nd(0):= 1:
cat4:= proc(a, b, c, d)
  ((a*10^nd(b)+b)*10^nd(c)+c)*10^nd(d)+d
end proc:
f:= proc(n) local L, i;
   L:= convert(n, base, 10);
   cat4(nops(L), seq(nops(select(t -> t mod 3 = i, L)), i=0..2))
end proc:
map(f, [$0..100]); # Robert Israel, Sep 07 2025

Mathematica

fc[n_, m_]:={Count[Mod[IntegerDigits[n], 3], m]}; a[n_]:=FromDigits[Join[{Length[IntegerDigits[n]]}, fc[n, 0], fc[n, 1], fc[n, 2]]]; Array[a, 37, 0] (* James C. McMahon, Sep 07 2025 *)

Prog

(Python)
def a(n):
    d, m = list(map(int, str(n))), [0, 0, 0]
    for di in d: m[di%3] += 1
    return int(str(len(d)) + "".join(map(str, m)))
print([a(n) for n in range(37)]) # Michael S. Branicky, Mar 28 2022

Crossrefs

Cf. A073053 (Sisyphus), A171797, A171798, A352752, A375208, A352751.

Sequence in context: A281039 A385708 A078199 * A271472 A147816 A050926

Adjacent sequences: A350706 A350707 A350708 * A350710 A350711 A350712

Keyword

nonn,base,easy

Author

Matthew E. Coppenbarger, Mar 28 2022

Status

approved