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Apply DENEAT operator (or the Sisyphus function) to n.
17

%I #48 Feb 16 2024 10:13:59

%S 101,11,101,11,101,11,101,11,101,11,112,22,112,22,112,22,112,22,112,

%T 22,202,112,202,112,202,112,202,112,202,112,112,22,112,22,112,22,112,

%U 22,112,22,202,112,202,112,202,112,202,112,202,112,112,22,112,22

%N Apply DENEAT operator (or the Sisyphus function) to n.

%C DENEAT(n): concatenate number of even digits in n, number of odd digits and total number of digits. E.g., 25 -> 1.1.2 = 112 (Digits: Even, Not Even, And Total). Leading zeros are then omitted.

%C This is also known as the Sisyphus function. - _N. J. A. Sloane_, Jun 25 2018

%C Repeated application of the DENEAT operator reduces all numbers to 123. This is easy to prove. Compare A073054, A100961. - _N. J. A. Sloane_ Jun 18 2005

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

%D M. Ecker, Caution: Black Holes at Work, New Scientist (Dec. 1992)

%D M. J. Halm, Blackholing, Mpossibilities 69, (Jan 01 1999), p. 2.

%D J. Schram, The Sisyphus string, J. Rec. Math., 19:1 (1987), 43-44.

%D M. Zeger, Fatal attraction, Mathematics and Computer Education, 27:2 (1993), 118-123.

%H N. J. A. Sloane, <a href="/A073053/b073053.txt">Table of n, a(n) for n = 0..20000</a>

%e a(1) = 0.1.1 -> 11.

%e a(10000000000) = 10111 because 10000000000 has 10 even digits, 1 odd digit and 11 total digits

%p read("transforms") :

%p A073053 := proc(n)

%p local e,o,L ;

%p if n = 0 then

%p 0 ;

%p else

%p e := A196563(n) ;

%p o := A196564(n) ;

%p L := [e,o,e+o] ;

%p digcatL(L) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jul 13 2012

%p # Maple code based on R. J. Mathar's code for A171797, added by _N. J. A. Sloane_, May 12 2019 (Start)

%p 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 cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:

%p 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 A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:

%p A171797 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1, n2, n1-n2]) ; end proc:

%p A073053 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2, n1-n2, n1]) ; end proc:

%p seq(A073053(n), n=1..80) ; (End)

%p L:=proc(n) if n=0 then 1 else floor(evalf(log(n)/log(10)))+1; fi; end;

%p S:=proc(n) local Le,Ld,Lt,t1,e,d,t; global L;

%p t1:=convert(n,base,10); e:=0; d:=0; t:=nops(t1);

%p for i from 1 to t do if (t1[i] mod 2) = 0 then e:=e+1; else d:=d+1; fi; od:

%p Le:=L(e); Ld:=L(d); Lt:=L(t);

%p if e=0 then 10^Lt*d+t

%p elif d=0 then 10^(Ld+Lt)*e+10^Lt*d+t

%p else 10^(Ld+Lt)*e+10^Lt*d+t; fi;

%p end;

%p [seq(S(n),n=1..200)]; # _N. J. A. Sloane_, Jun 25 2018

%p # alternative Maple program:

%p a:= n-> (l-> (e-> parse(cat(e, (h-> [h-e, h][])(nops(l))))

%p )(nops(select(x-> x::even, l))))(convert(n, base, 10)):

%p seq(a(n), n=0..200); # _Alois P. Heinz_, Jan 21 2022

%t f[n_] := Block[{id = IntegerDigits[n]}, FromDigits[ Join[ IntegerDigits[ Length[ Select[id, EvenQ[ # ] &]]], IntegerDigits[ Length[ Select[id, OddQ[ # ] &]]], IntegerDigits[ Length[ id]] ]]]; Table[ f[n], {n, 0, 55}] (* _Robert G. Wilson v_, Jun 09 2005 *)

%t s={};Do[id=IntegerDigits[n];ev=Select[id, EvenQ];ne=Select[id, OddQ];fd=FromDigits[{Length[ev], Length[ne], Length[id]}]; s=Append[s, fd], {n, 81}];SameQ[newA073053-s] (* _Zak Seidov_ *)

%t deneat[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Flatten[ IntegerDigits/@ {Count[ idn,_?EvenQ],Count[ idn,_?OddQ],Length[ idn]}]]] Array[ deneat,60,0]// Flatten (* _Harvey P. Dale_, Aug 13 2021 *)

%o (Python)

%o def a(n):

%o s = str(n)

%o e = sum(1 for c in s if c in "02468")

%o return int(str(e) + str(len(s)-e) + str(len(s)))

%o print([a(n) for n in range(54)]) # _Michael S. Branicky_, Jan 21 2022

%Y Cf. A008577, A072420, A073054, A100961, A171797.

%Y For records see A305395, A004643, A308004.

%K easy,nonn,base

%O 0,1

%A _Michael Joseph Halm_, Aug 16 2002

%E Edited and corrected by _Jason Earls_ and _Robert G. Wilson v_, Jun 03 2005

%E a(0) added by _N. J. A. Sloane_, May 12 2019