A196563 -id:A196563 - cp's OEIS frontend

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

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

A102679 Number of digits >= 7 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007093 (numbers in base 7). - Bernard Schott, Feb 12 2023

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007093, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102680.

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=7 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..125); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 3/10) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(7*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A104639 Number of even digits in n^3.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 1, 1, 3, 0, 2, 1, 3, 0, 3, 1, 2, 2, 4, 2, 4, 2, 3, 2, 1, 2, 2, 3, 4, 1, 3, 0, 2, 3, 4, 2, 3, 0, 5, 3, 4, 1, 3, 1, 1, 3, 2, 2, 4, 2, 5, 3, 3, 2, 2, 1, 1, 2, 5, 4, 4, 4, 5, 4, 4, 3, 3, 3, 4, 0, 3, 2, 5, 3, 3, 2, 3, 2, 4, 2, 2, 1, 3, 3, 4, 3, 4, 3, 4, 0, 4, 3, 4, 1, 4, 2, 2, 2, 6

Offset: 1

Zak Seidov, Mar 18 2005

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A104640, A196563.

Table[Count[IntegerDigits[n^3], ?EvenQ], {n, 100}] (* _G. c. Greubel, Aug 13 2018 *)

a(n) = my(d = digits(n^3)); sum(i=1, #d, 1 - (d[i] % 2)); \\ Michel Marcus, Oct 05 2013

a(n) = A196563(n^3). - Michel Marcus, Oct 05 2013

A065031 In the decimal expansion of n, replace each odd digit with 1 and each even digit with 2.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 22, 21, 22, 21, 22, 21, 22, 21, 22, 21

Offset: 0

Santi Spadaro, Nov 03 2001

A196563(a(n)) = A196563(n); A196564(a(n)) = A196564(n).

			a(123)=121 because 1 and 3 are odd and 2 is even.

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

a065031 n = f n  where
   f x | x < 10    = 2 - x `mod` 2
       | otherwise = 10 * (f x') + 2 - m `mod` 2
       where (x',m) = divMod x 10
-- Reinhard Zumkeller, Feb 22 2012

Table[FromDigits[If[OddQ[#],1,2]&/@IntegerDigits[n]],{n,0,120}] (* Harvey P. Dale, Jun 08 2014 *)

A102681 Number of digits >= 8 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007094 (numbers in base 8). - Bernard Schott, Feb 18 2023

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007094, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102682.

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=8 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..120); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 1/5) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(8*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A338742 When a(n) is even, a(n) is the number of even digits present so far in the sequence, a(n) included.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 2, 23, 4, 25, 6, 27, 8, 29, 10, 31, 33, 35, 37, 39, 41, 12, 43, 14, 45, 16, 47, 18, 20, 22, 24, 26, 28, 49, 30, 51, 53, 55, 57, 59, 61, 32, 63, 34, 65, 36, 67, 38, 40, 42, 44, 46, 48, 69, 50, 71, 73, 75, 77, 79, 81, 52, 83, 54, 85, 56, 87, 58, 60, 62, 64, 66, 68, 89

Offset: 1

Eric Angelini and Carole Dubois, Nov 06 2020

The odd nonnegative integers are present in their natural order. Some even natural integers will never appear.

			The first even term is a(12) = 2 and there are indeed 2 even digits so far in the sequence (the 2 from 21 and 2 itself);
The next even term is a(14) = 4 and there are now 4 even digits so far (2, 2, 2 and 4);
The next even term is a(16) = 6 and there are now 6 even digits so far (2, 2, 2, 4, 2 and 6); etc.

Cf. A338741, A338743, A338744, A338745, A338746 (variants on the same idea), A196563.

Block[{a = {0}, c = 0}, Do[Block[{k = 1, s}, While[If[EvenQ[k], Nand[FreeQ[a, k], k == c + Set[s, Total@ DigitCount[k, 10, {0, 2, 4, 6, 8}]]], ! FreeQ[a, k]], k++]; If[EvenQ[k], c += s, c += Total@ DigitCount[k, 10, {0, 2, 4, 6, 8}]]; AppendTo[a, k]], {i, 79}]; a] (* Michael De Vlieger, Nov 06 2020 *)

A338743 When a(n) is odd, a(n) is the number of even digits present so far in the sequence, a(n) included.

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, 22, 24, 26, 28, 21, 30, 23, 32, 25, 34, 27, 36, 29, 38, 40, 42, 44, 46, 48, 41, 50, 43, 52, 45, 54, 47, 56, 49, 58, 60, 62, 64, 66, 68, 61, 70, 63, 72, 65, 74, 67, 76, 69, 78, 80, 82, 84, 86, 88, 81, 90, 83, 92, 85, 94, 87, 96, 89, 98, 100, 102, 104

Offset: 1

Eric Angelini and Carole Dubois, Nov 06 2020

The even nonnegative integers are present in their natural order. Some odd natural integers will never appear (11 for instance).

			The first odd term is a(2) = 1 and there is indeed 1 even digit so far in the sequence (0);
The next odd term is a(5) = 3 and there are now 3 even digits so far (0, 2 and 4);
The next odd term is a(8) = 5 and there are now 5 even digits so far (0, 2, 4, 6 and 8);
...
The term a(21) = 21 and there are indeed 21 even digits in the sequence so far (0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 2, 0, 2, 2, 2, 4, 2, 6, 2, 8, 2); etc.

Cf. A338741, A338742, A338744, A338745, A338746 (variants on the same idea), A196563.

Block[{a = {0}, c = 1}, Do[Block[{k = 1, s}, While[If[OddQ[k], Nand[FreeQ[a, k], k == c + Set[s, Total@DigitCount[k, 10, {0, 2, 4, 6, 8}]]], ! FreeQ[a, k]], k++]; If[OddQ[k], c += s, c += Total@ DigitCount[k, 10, {0, 2, 4, 6, 8}]]; AppendTo[a, k]], {i, 77}]; a] (* Michael De Vlieger, Nov 06 2020 *)

A102677 Number of digits >= 6 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007092 (numbers in base 6). - Bernard Schott, Feb 02 2023

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007092, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102678.

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=6 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..116); # Emeric Deutsch, Feb 23 2005

Table[Total@ Take[Most@ DigitCount@ n, -4], {n, 0, 104}] (* Michael De Vlieger, Aug 17 2017 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 2/5) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(6*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A352547 Numbers having more odd than even digits when written in base 10.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, 35, 37, 39, 51, 53, 55, 57, 59, 71, 73, 75, 77, 79, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 123, 125, 127, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155

Offset: 1

M. F. Hasler, Jul 03 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A196563, A196564.
Cf. A072600 (same in base 2).
Cf. A227870, A352546 (numbers with fewer odd than even decimal digits).

A352547Q[k_] := Length[#] < 2*Count[#, _?OddQ] &[IntegerDigits[k]];
Select[Range[300], A352547Q] (* Paolo Xausa, Nov 28 2024 *)

select( {is_A352547(n)=vecsum(n=digits(n)%2)*2>#n}, [0..155])

def ok(n): return len(s:=str(n)) > 2*sum(1 for c in s if c in "02468")
print([k for k in range(156) if ok(k)]) # Michael S. Branicky, Jul 03 2022

A007928 Numbers containing an even digit.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 98, 100

Offset: 1

R. Muller

Or, numbers whose product of digits is even.

Complement of A014261; A196563(a(n)) > 0. - Reinhard Zumkeller, Oct 04 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Index entries for 10-automatic sequences.

import Data.List (findIndices)
a007928 n = a007928_list !! (n-1)
a007928_list = findIndices (> 0) a196563_list
-- Reinhard Zumkeller, Oct 04 2011

[ n : n in [0..129] | IsEven(&*Intseq(n,10)) ];

is(n)=vecmin(Set(digits(n)%2))==0 \\ Charles R Greathouse IV, Feb 14 2017

a(n) ~ n. - Charles R Greathouse IV, Sep 07 2012

cp's OEIS Frontend

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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A102679 Number of digits >= 7 in decimal representation of n.

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A104639 Number of even digits in n^3.

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A065031 In the decimal expansion of n, replace each odd digit with 1 and each even digit with 2.

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A102681 Number of digits >= 8 in decimal representation of n.

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A338742 When a(n) is even, a(n) is the number of even digits present so far in the sequence, a(n) included.

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A338743 When a(n) is odd, a(n) is the number of even digits present so far in the sequence, a(n) included.

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A102677 Number of digits >= 6 in decimal representation of n.

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