A073053 -id:A073053 - cp's OEIS frontend

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

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

A308104 Distinct values taken by the DENEAT operator (A073053) in increasing order.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1010

Offset: 1

Rémy Sigrist, May 13 2019

This sequence corresponds to numbers of the form e.o.(e+o) with e, o >= 0 and e+o > 0 (where "." denotes concatenation in decimal).

This sequence first differs from A108203 for n = 55: a(55) = 1010 whereas A108203(55) = 1001.

Rémy Sigrist, Scatterplot of the first difference of the first 188281 terms (the terms < 100000000)
Rémy Sigrist, PARI program for A308104

See A308106 for the values in order of appearance.

Cf. A073053, A108203.

PARI
```
See Links section.
```

A305395 Records in A073053.

Original entry on oeis.org

11, 101, 112, 202, 213, 303, 314, 404, 415, 505

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. A073053, A004643.

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 *)

A308106 Distinct values taken by the DENEAT operator (A073053) in order of appearance.

Original entry on oeis.org

101, 11, 112, 22, 202, 213, 123, 33, 303, 314, 224, 134, 44, 404, 415, 325, 235, 145, 55, 505, 516, 426, 336, 246, 156, 66, 606, 617, 527, 437, 347, 257, 167, 77, 707, 718, 628, 538, 448, 358, 268, 178, 88, 808, 819, 729, 639, 549, 459, 369, 279, 189, 99, 909

Offset: 1

Rémy Sigrist, May 13 2019

			A073053 starts:       101, 11, 101, 11, 101, 11, 101, 11, 101, 11, 112, 22, ...
This sequence starts: 101, 11,                                     112, 22, ...

Rémy Sigrist, PARI program for A308106

See A308104 for the values in increasing order.

Cf. A073053.

PARI
```
See Links section.
```

A173578 a(n) = a(n-1) + A073053(a(n-1)).

Original entry on oeis.org

1, 12, 124, 337, 370, 493, 616, 829, 1042, 1356, 1490, 1714, 1848, 2162, 2476, 2790, 3014, 3238, 3462, 3776, 3910, 4044, 4448, 4852, 5166, 5390, 5524, 5748, 5972, 6106, 6420, 6824, 7228, 7542, 7766, 7990, 8124, 8438, 8752, 8976, 9200, 9514, 9648, 9962

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Mar 05 2010

A073053 sequence is operator DENEAT: concatenate number of even digits in n, number of odd digits and total number of digits.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A173578 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        procname(n-1)+A073053(procname(n-1)) ;
    end if;
end proc:
seq(A173578(n),n=1..20) ; # R. J. Mathar, Jul 13 2012

deneat[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[ Flatten[ IntegerDigits/@ {Count[ idn,?EvenQ],Count[ idn,?OddQ],Length[ idn]}]]]; NestList[ #+deneat[ #]&,1,50] (* Harvey P. Dale, Aug 13 2021 *)

a(n) = a(n-1)+ A073053(a(n-1)).

A308125 Numbers k that are a multiple of the DENEAT operator applied to k (A073053).

Original entry on oeis.org

0, 22, 44, 66, 88, 123, 264, 369, 462, 615, 660, 738, 759, 852, 957, 1120, 1344, 1568, 1884, 2024, 2068, 2200, 2244, 2288, 2420, 2464, 2640, 2684, 2860, 2912, 3350, 3360, 3584, 3752, 4004, 4048, 4224, 4268, 4400, 4444, 4488, 4620, 4664, 4840, 4884, 5024, 6028, 6204

Offset: 0

Paolo P. Lava, May 14 2019

The DENEAT operator is also known as the Sisyphus function.

On the other hand, the sequence of numbers k such that DENEAT(k) is a multiple of k, is the finite sequence {1, 11, 14, 16, 22, 56, 123}.

			2912 / DENEAT(2912) = 2912 / 224 = 13.

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

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A073053, A073054.

P:=proc(n) local a,b,c,d,k; a:=convert(n,base,10); b:=0: c:=0:
for k from 1 to nops(a) do if a[k] mod 2=0 then b:=b+1; else c:=c+1; fi;
od: d:=b*10^length(c)+c; a:=n/(d*10^length(length(n))+length(n)):
if frac(a)=0 then n; fi; end: 0,seq(P(i),i=1..6204);

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

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Extensions

A308104 Distinct values taken by the DENEAT operator (A073053) in increasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A305395 Records in A073053.

Views

Author

Keywords

Comments

References

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A308106 Distinct values taken by the DENEAT operator (A073053) in order of appearance.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A173578 a(n) = a(n-1) + A073053(a(n-1)).

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Formula

A308125 Numbers k that are a multiple of the DENEAT operator applied to k (A073053).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Python