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A308005 A modified Sisyphus function: a(n) = concatenation of (number of odd digits in n) (number of digits in n) (number of even digits in n).

%I A308005 #16 May 14 2019 08:57:36
%S A308005 11,110,11,110,11,110,11,110,11,110,121,220,121,220,121,220,121,220,
%T A308005 121,220,22,121,22,121,22,121,22,121,22,121,121,220,121,220,121,220,
%U A308005 121,220,121,220,22,121,22,121,22,121,22,121,22,121,121,220,121,220,121,220,121,220,121,220,22,121,22,121,22
%N A308005 A modified Sisyphus function: a(n) = concatenation of (number of odd digits in n) (number of digits in n) (number of even digits in n).
%C A308005 If we start with n and repeatedly apply the map i -> a(i), it appears that we eventually reach one of the two fixed points 22 or 231, or enter the two-cycle (33, 220). Are there any other possibilities? This is in contrast to the behavior of the closely related A308003.
%D A308005 M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.
%H A308005 Robert Israel, <a href="/A308005/b308005.txt">Table of n, a(n) for n = 0..10000</a>
%e A308005 11 has 2 digits, both odd, so a(11)=220.
%e A308005 12 has 2 digits, one even and one odd, so a(12)=121. Then a(121) = 231, a fixed point.
%e A308005 22 has two digits, both even, so 22 -> 22, another fixed point  (leading zeros are omitted).
%p A308005 Maple code based on _R. J. Mathar_'s code for A171797:
%p A308005 nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n, base, 10) do if type(d, 'even') then a :=a +1 ; end if; end do; a ; end proc:
%p A308005 cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:
%p A308005 catL := proc(L) local a, i; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:
%p A308005 A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:
%p A308005 A308005 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1-n2, n1, n2]) ; end proc:
%p A308005 [seq(A308005(n), n=0..80)];
%Y A308005 Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A308003.
%K A308005 nonn,base,easy
%O A308005 0,1
%A A308005 _N. J. A. Sloane_, May 12 2019