A171797 -id:A171797 - cp's OEIS frontend

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

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.

A100961 For a decimal string s, let f(s) = decimal string ijk, where i = number of even digits in s, j = number of odd digits in s, k=i+j (see A171797). Start with s = decimal expansion of n; a(n) = number of applications of f needed to reach the string 123.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane, Jun 17 2005

Obviously if the digits of m and n have the same parity then a(m) = a(n). E.g. a(334) = a(110). In other words, a(n) = a(A065031(n)).

It is easy to show that (i) the trajectory of every number under f eventually reaches 123 (if s has more than three digits then f(s) has fewer digits than s) and (ii) since each string ijk has only finitely many preimages, a(n) is unbounded.

			n=0: s=0 -> f(s) = 101 -> f(f(s)) = 123, stop, a(0) = 2.
n=1: s=1 => f(s) = 011 -> f(f(s)) = 123, stop, f(1) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

A073054 gives another version. f(n) is (essentially) A171797 or A073053.

Cf. A065031, A308002.

More terms from Zak Seidov, Jun 18 2005

A305396 Records in A171797.

Original entry on oeis.org

101, 110, 211, 220, 321, 330, 431, 440, 541, 550

Offset: 1

N. J. A. Sloane, Jun 25 2018

The record-holders are the powers of 2 written in base 4, A004643.

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

Cf. A171797, A004643.

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

Original entry on oeis.org

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

A171798 a(n) = base-10 concatenation XYZ, where X = number of bits in binary expansion of n, Y = number of 0's, Z = number of 1's.

Original entry on oeis.org

101, 211, 202, 321, 312, 312, 303, 431, 422, 422, 413, 422, 413, 413, 404, 541, 532, 532, 523, 532, 523, 523, 514, 532, 523, 523, 514, 523, 514, 514, 505, 651, 642, 642, 633, 642, 633, 633, 624, 642, 633, 633, 624, 633, 624, 624, 615, 642, 633, 633, 624

Offset: 1

N. J. A. Sloane, Oct 15 2010, Oct 16 2010

Start with n, repeatedly apply the map i -> a(i). Then every n converges to one of 1019, 1147, 1165 or 14311 (cf. A171813). Proof: this is true by direct calculation for n=1..2^14. For larger n, a(n) < n.

			14 = 1110 in base 2, so X=4, Y=1, Z=3, a(14)=413.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A171797, A171813, A171823, A171825.

Cf. A070939, A023416, A000120.

a171798 n = read $ concatMap (show . ($ n))
                   [a070939, a023416, a000120] :: Integer
-- Reinhard Zumkeller, Feb 22 2012

# Maple code for trajectories of numbers from 1 to M:
F:=proc(n) local s,t1,t2; t1:=convert(n,base,2); t2:=nops(t1); s:=add(t1[i],i=1..t2);
parse(cat(t2,t2-s,s)); end;
M:=16384;
for n from 1 to M do t3:=F(n); sw:=-1;
for i from 1 to 10 do
if (t3 = 1147) or (t3 = 1165) or (t3 = 1019) or (t3 = 14311) then sw:=1; break; fi;
t3:=F(t3);
od;
if sw < 0 then lprint(n); fi;
od:
Contribution from R. J. Mathar, Oct 15 2010: (Start)
read("transforms") ; cat2 := proc(a,b) dgsb := max(1,ilog10(b)+1) ; a*10^dgsb+b ; end proc:
catL := proc(L) local a; a := op(1,L) ; for i from 2 to nops(L) do a := cat2(a,op(i,L)) ; end do; a; end proc:
A070939 := proc(n) max(1,ilog2(n)+1) ; end proc:
A171798 := proc(n) local n1,n3 ; n1 := A070939(n) ; n3 := wt(n) ; catL([n1,n1-n3,n3]) ; end proc:
seq(A171798(n),n=1..80) ; (End)

ans[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[{Length[idn2],Count[idn2,0],Count[idn2,1]}]]; Table[ans[i], {i, 50}] (* Harvey P. Dale, Nov 06 2010 *)

def a(n):
    b = bin(n)[2:]
    z = b.count("0")
    return int(str(len(b)) + str(z) + str(len(b)-z))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Mar 28 2022

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

A308003 A modified Sisyphus function: a(n) = concatenation of (number of even digits in n) (number of digits in n) (number of odd digits in n).

Original entry on oeis.org

110, 11, 110, 11, 110, 11, 110, 11, 110, 11, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 220, 121, 220, 121, 220, 121, 220, 121, 220, 121, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 220, 121, 220, 121, 220, 121, 220, 121, 220, 121, 121, 22, 121, 22, 121

Offset: 0

N. J. A. Sloane, May 12 2019

If we start with n and repeatedly apply the map i -> a(i), we eventually reach 132 (see A073054).

			11 has 2 digits, both odd, so a(11)=22 (leading zeros are omitted).
12 has 2 digits, one even and one odd, so a(12)=121. Then a(121) = 132.

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

N. J. A. Sloane, Table of n, a(n) for n = 0..28000

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002.

A073054 gives steps to reach 132.

# Maple code based on R. J. Mathar's code for A171797:
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:
A308003 := proc(n) local n1,n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2,n1,n1-n2]) ; end proc:
seq(A308003(n),n=0..80) ;

def a(n):
    s = str(n)
    e = sum(1 for c in s if c in "02468")
    return int(str(e) + str(len(s)) + str(len(s)-e))
print([a(n) for n in range(55)]) # Michael S. Branicky, Mar 29 2022

A350709 Modified Sisyphus function of order 3: a(n) is the concatenation of (number of digits of n)(number digits of n congruent to 0 modulo 3)(number of digits of n congruent to 1 modulo 3)(number of digits of n congruent to 2 modulo 3).

Original entry on oeis.org

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

Matthew E. Coppenbarger, Mar 28 2022

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

			11 has two digits, both congruent to 1 modulo 3, so a(11) = 2020.
a(20) = 2101.
a(30) = 2200.
a(1111123567) = 10262.

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

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002.

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

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 *)

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).

Original entry on oeis.org

11, 110, 11, 110, 11, 110, 11, 110, 11, 110, 121, 220, 121, 220, 121, 220, 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, 121, 22, 121, 22, 121, 121, 220, 121, 220, 121, 220, 121, 220, 121, 220, 22, 121, 22, 121, 22

Offset: 0

N. J. A. Sloane, May 12 2019

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.

			11 has 2 digits, both odd, so a(11)=220.
12 has 2 digits, one even and one odd, so a(12)=121. Then a(121) = 231, a fixed point.
22 has two digits, both even, so 22 -> 22, another fixed point  (leading zeros are omitted).

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

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A308003.

Maple code based on R. J. Mathar's code for A171797:
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:
A308005 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1-n2, n1, n2]) ; end proc:
[seq(A308005(n), n=0..80)];

A375208 Modified Sisyphus function of order 5.

Original entry on oeis.org

110000, 101000, 100100, 100010, 100001, 110000, 101000, 100100, 100010, 100001, 211000, 202000, 201100, 201010, 201001, 211000, 202000, 201100, 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

Offset: 0

Matt Coppenbarger, Oct 16 2024

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.

If we start with n and repeatedly apply the map i -> a(i), we eventually get the cycle {613200, 622110}.

			11 has two digits, both congruent to 1 modulo 5, so a(11) = 202000.
a(20) = 210100.
a(30) = 210010.
a(2527200000) = 1060400.

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

Cf. A073053 (Sisyphus), A171797, A171798, A171813, A055642, A196563, A196564, A308002, A350709.

a:= n-> (l-> parse(cat(nops(l), seq(add(`if`(irem(i, 5)=k
          , 1, 0), i=l), k=0..4))))(convert(n, base, 10)):
seq(a(n), n=0..44);  # Alois P. Heinz, Oct 23 2024

# based on Michael S. Branicky in A350709
def a(n, order=5):
    d, m = list(map(int, str(n))), [0]*order
    for di in d: m[di%order] += 1
    return int(str(len(d)) + "".join(map(str, m)))
print([a(n) for n in range(37)])

from collections import Counter
def A375208(n):
    s = str(n)
    c = Counter(int(d)%5 for d in s)
    return int(str(len(s))+''.join(str(c[i]) for i in range(5))) # Chai Wah Wu, Nov 26 2024

cp's OEIS Frontend

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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A100961 For a decimal string s, let f(s) = decimal string ijk, where i = number of even digits in s, j = number of odd digits in s, k=i+j (see A171797). Start with s = decimal expansion of n; a(n) = number of applications of f needed to reach the string 123.

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A305396 Records in A171797.

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

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A171798 a(n) = base-10 concatenation XYZ, where X = number of bits in binary expansion of n, Y = number of 0's, Z = number of 1's.

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A308003 A modified Sisyphus function: a(n) = concatenation of (number of even digits in n) (number of digits in n) (number of odd digits in n).

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A350709 Modified Sisyphus function of order 3: a(n) is the concatenation of (number of digits of n)(number digits of n congruent to 0 modulo 3)(number of digits of n congruent to 1 modulo 3)(number of digits of n congruent to 2 modulo 3).

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

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