A100961 -id:A100961 - cp's OEIS frontend

A073053 Apply DENEAT operator (or the Sisyphus function) to n.

101, 11, 101, 11, 101, 11, 101, 11, 101, 11, 112, 22, 112, 22, 112, 22, 112, 22, 112, 22, 202, 112, 202, 112, 202, 112, 202, 112, 202, 112, 112, 22, 112, 22, 112, 22, 112, 22, 112, 22, 202, 112, 202, 112, 202, 112, 202, 112, 202, 112, 112, 22, 112, 22

Offset: 0

Michael Joseph Halm, Aug 16 2002

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.
This is also known as the Sisyphus function. - N. J. A. Sloane, Jun 25 2018
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

			a(1) = 0.1.1 -> 11.
a(10000000000) = 10111 because 10000000000 has 10 even digits, 1 odd digit and 11 total digits

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.
M. Ecker, Caution: Black Holes at Work, New Scientist (Dec. 1992)
M. J. Halm, Blackholing, Mpossibilities 69, (Jan 01 1999), p. 2.
J. Schram, The Sisyphus string, J. Rec. Math., 19:1 (1987), 43-44.
M. Zeger, Fatal attraction, Mathematics and Computer Education, 27:2 (1993), 118-123.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

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

For records see A305395, A004643, A308004.

read("transforms") :
A073053 := proc(n)
    local e,o,L ;
    if n = 0 then
        0 ;
    else
        e := A196563(n) ;
        o := A196564(n) ;
        L := [e,o,e+o] ;
        digcatL(L) ;
    end if;
end proc: # R. J. Mathar, Jul 13 2012
# Maple code based on R. J. Mathar's code for A171797, added by N. J. A. Sloane, May 12 2019 (Start)
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:
cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:
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:
A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:
A171797 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1, n2, n1-n2]) ; end proc:
A073053 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2, n1-n2, n1]) ; end proc:
seq(A073053(n), n=1..80) ; (End)
L:=proc(n) if n=0 then 1 else floor(evalf(log(n)/log(10)))+1; fi; end;
S:=proc(n) local Le,Ld,Lt,t1,e,d,t; global L;
t1:=convert(n,base,10); e:=0; d:=0; t:=nops(t1);
for i from 1 to t do if (t1[i] mod 2) = 0 then e:=e+1; else d:=d+1; fi; od:
Le:=L(e); Ld:=L(d); Lt:=L(t);
if e=0 then 10^Lt*d+t
elif d=0 then 10^(Ld+Lt)*e+10^Lt*d+t
else 10^(Ld+Lt)*e+10^Lt*d+t; fi;
end;
[seq(S(n),n=1..200)]; # N. J. A. Sloane, Jun 25 2018
# alternative Maple program:
a:= n-> (l-> (e-> parse(cat(e, (h-> [h-e, h][])(nops(l))))
    )(nops(select(x-> x::even, l))))(convert(n, base, 10)):
seq(a(n), n=0..200);  # Alois P. Heinz, Jan 21 2022

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 *)
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 *)
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 *)

def a(n):
    s = str(n)
    e = sum(1 for c in s if c in "02468")
    return int(str(e) + str(len(s)-e) + str(len(s)))
print([a(n) for n in range(54)]) # Michael S. Branicky, Jan 21 2022

Edited and corrected by Jason Earls and Robert G. Wilson v, Jun 03 2005

a(0) added by N. J. A. Sloane, May 12 2019

A171797 A modified Sisyphus function: a(n) = concatenation of (number of digits in n) (number of even digits) (number of odd digits).

Original entry on oeis.org

110, 101, 110, 101, 110, 101, 110, 101, 110, 101, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 211, 220, 211, 220, 211, 220, 211, 220, 211, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 211, 220, 211, 220, 211, 220, 211, 220, 211, 211, 202

Offset: 0

N. J. A. Sloane, Oct 15 2010

Start with n, repeatedly apply the map i -> a(i). Then every number converges to 312. - Eric Angelini and Alexandre Wajnberg, Oct 15 2010

			11 has 2 digits, both odd, so a(11) = 202.
12 has 2 digits, one even and one odd, so a(12)=211. Then a(211) = 312.

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..25000

Cf. A073053 (Sisyphus), A171798, A171813, A055642, A196563, A196564, A308002, A308003 (another version).

A100961 gives steps to reach 312.

a171797 n = read $ concatMap (show . ($ n))
                   [a055642, a196563, a196564] :: Integer
-- Reinhard Zumkeller, Feb 22 2012, Oct 15 2010

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:
cat2 := proc(a,b) local ndigsb; ndigsb := max(ilog10(b)+1,1) ; a*10^ndigsb+b ; end:
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:
A055642 := proc(n) max(1,ilog10(n)+1) ; end proc:
A171797 := proc(n) local n1,n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1,n2,n1-n2]) ; end proc:
seq(A171797(n),n=1..80) ; # R. J. Mathar, Oct 15 2010 and Oct 18 2010

def a(n):
    s = str(n); e = sum(d in "02468" for d in s)
    return int("".join(map(str, (len(s), e, len(s)-e))))
print([a(n) for n in range(52)]) # Michael S. Branicky, Jun 15 2021

More terms from R. J. Mathar, Oct 15 2010

a(0) added by N. J. A. Sloane, May 12 2019

A308002 a(n) = smallest nonnegative number that requires n applications of the modified Sisyphus function x -> A171797(x) to reach 312.

Original entry on oeis.org

312, 101, 0, 11, 10000000000111111111

Offset: 0

N. J. A. Sloane, May 12 2019

The next term, a(5), is 1 0^1000099 1^100010, a number with 1100110 digits, and is too large to display here.

a(n) = index of first n in A100961.

			0 -> 110 -> 312 reaches 312 in two steps, so a(2) = 0.

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

Cf. A100961, A171797.

A073054 Number of applications of DENEAT operator x -> A073053(x) needed to transform n to 123.

Original entry on oeis.org

2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4

Offset: 0

Michael Joseph Halm, Aug 16 2002

0 first occurs for n=123, 1 first occurs for n=101, 2 first occurs for n=0, 3 first occurs for n=20, 4 first occurs for n=11, 5 first occurs for n=1. What is the least n such that a(n) > 5? - Jason Earls, Jun 03 2005 (Corrected by N. J. A. Sloane, May 12 2019)

Since each string has only finitely many preimages under this map, the sequence is unbounded. Compare A100961. - N. J. A. Sloane, Jun 18 2005

M. Ecker, Caution: Black Holes at Work, New Scientist (Dec. 1992)
M. J. Halm, Blackholing, Mpossibilities 69, (1999), p. 2.

Cf. A073053, A100961.

f[n_] := Block[{id = IntegerDigits[n]}, FromDigits[ Join[ IntegerDigits[ Length[ Select[id, EvenQ[ # ] &]]], IntegerDigits[ Length[ Select[id, OddQ[ # ] &]]], IntegerDigits[ Length[ id]] ]]]; Table[ Length[ NestWhileList[f, n, UnsameQ, All]] - 2, {n, 0, 104}] (* Robert G. Wilson v, Jun 09 2005 *)

Edited and corrected by Jason Earls and Robert G. Wilson v, Jun 03 2005

Offset corrected by N. J. A. Sloane, May 12 2019

A308004 a(n) = smallest nonnegative number that requires n applications of the Sisyphus function x -> A073053(x) to reach 123.

Original entry on oeis.org

123, 101, 0, 20, 11, 1

Offset: 0

N. J. A. Sloane, May 12 2019

a(n) = index of first n in A073054.

a(6) is currently unknown.

			0 -> 101 -> 123 reaches 123 in two steps, so a(2) = 0.
1 -> 11 -> 22 -> 202 -> 303 -> 123 reaches 123 in 5 steps, so a(5) = 1.

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

Cf. A073053, A073054, A100961, A171797.

id[n_]:=IntegerDigits[n]; il[n_]:=If[n!=0,IntegerLength[n],1]
den[n_]:=FromDigits[{Length[Select[id[n],EvenQ]],Length[Select[id[n],OddQ]],il[n]}]; numD[n_]:=Length[FixedPointList[den,n]]-2;
a308004[n_]:=Module[{k=0},While[numD[k]!=n,k++];k];
a308004/@Range[0,5] (* Ivan N. Ianakiev, May 13 2019 *)

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A073053 Apply DENEAT operator (or the Sisyphus function) to n.

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A171797 A modified Sisyphus function: a(n) = concatenation of (number of digits in n) (number of even digits) (number of odd digits).

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A308002 a(n) = smallest nonnegative number that requires n applications of the modified Sisyphus function x -> A171797(x) to reach 312.

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A073054 Number of applications of DENEAT operator x -> A073053(x) needed to transform n to 123.

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Programs

Mathematica

Extensions

A308004 a(n) = smallest nonnegative number that requires n applications of the Sisyphus function x -> A073053(x) to reach 123.

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Mathematica