A061217 -id:A061217 - cp's OEIS frontend

A196564 Number of odd digits in decimal representation of n.

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

Offset: 0

Reinhard Zumkeller, Oct 04 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Zachary P. Bradshaw and Christophe Vignat, Dubious Identities: A Visit to the Borwein Zoo, arXiv:2307.05565 [math.HO], 2023.

Cf. A014261, A014263, A027868, A046034, A055640, A055641, A055642, A061217, A102669-A102685, A122640, A196563.

a196564 n = length [d | d <- show n, d `elem` "13579"]
-- Reinhard Zumkeller, Feb 22 2012, Oct 04 2011

A196564 := proc(n)
        if n =0 then
                0;
        else
                convert(n,base,10) ;
                add(d mod 2,d=%) ;
        end if:
end proc: # R. J. Mathar, Jul 13 2012

Table[Total[Mod[IntegerDigits[n],2]],{n,0,100}] (* Zak Seidov, Oct 13 2015 *)

a(n) = #select(x->x%2, digits(n)); \\ Michel Marcus, Oct 14 2015

def a(n): return sum(1 for d in str(n) if d in "13579")
print([a(n) for n in range(100)]) # Michael S. Branicky, May 15 2022

a(n) = A055642(n) - A196563(n);
a(A014263(n)) = 0; a(A007957(n)) > 0.
From Hieronymus Fischer, May 30 2012: (Start)
a(n) = Sum_{j=0..m} (floor(n/(2*10^j) + (1/2)) - floor(n/(2*10^j))), where m=floor(log_10(n)).
a(10*n+k) = a(n) + a(k), 0<=k<10, n>=0.
a(n) = a(floor(n/10)) + a(n mod 10), n>=0.
a(n) = Sum_{j=0..m} a(floor(n/10^j) mod 10), n>=0.
a(A014261(n)) = floor(log_5(4*n+1)), n>0.
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} x^10^j/(1+x^10^j). (End)

A196563 Number of even digits in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Oct 04 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Zachary P. Bradshaw and Christophe Vignat, Dubious Identities: A Visit to the Borwein Zoo, arXiv:2307.05565 [math.HO], 2023.

Cf. A014261, A014263, A027868, A046034, A055640, A055641, A055642, A061217, A102669-A102685, A122640, A196564.

a196563 n = length [d | d <- show n, d `elem` "02468"]
-- Reinhard Zumkeller, Feb 22 2012, Oct 04 2011

A196563 := proc(n)
        if n =0 then
                1;
        else
                convert(n,base,10) ;
                add(1-(d mod 2),d=%) ;
        end if:
end proc: # R. J. Mathar, Jul 13 2012

Table[Count[Mod[IntegerDigits[n],2],0][n],{n,0,100}] (* Zak Seidov, Oct 13 2015 *)
Table[Count[IntegerDigits[n],?EvenQ],{n,0,120}] (* _Harvey P. Dale, Feb 22 2020 *)

a(n) = #select(x->(!(x%2)), if (n, digits(n), [0])); \\ Michel Marcus, Oct 14 2015

def a(n): return sum(1 for d in str(n) if d in "02468")
print([a(n) for n in range(100)]) # Michael S. Branicky, May 15 2022

a(n) = A055642(n) - A196564(n);
a(A014261(n)) = 0; a(A007928(n)) > 0.
From Hieronymus Fischer, May 30 2012: (Start)
a(n) = Sum_{j=0..m} (1 + floor(n/(2*10^j)) - floor(n/(2*10^j) + (1/2))), where m=floor(log_10(n)).
a(10*n+k) = a(n) + a(k), 0<=k<10, n>=1.
a(n) = a(floor(n/10))+a(n mod 10), n>=10.
a(n) = Sum_{j=0..m} a(floor(n/10^j) mod 10), n>=0.
a(A014263(n)) = 1 + floor(log_5(n-1)), n>1.
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} x^(2*10^j)/(1+x^10^j). (End)

A083449 a(n) = A019566(n)/9, where A019566(n) = concat(n,...,1) - concat(1,...,n).

Original entry on oeis.org

0, 1, 22, 343, 4664, 58985, 713306, 8367627, 96021948, -150891621, -13731137410, -260644605199, 86159119727012, 19839246664059223, 3106259112208391434, 422859356777752723645, 53509280234443297055856, 6473262479112108841388067, 759559693477989774385720278

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 01 2003

Are there any palindromes > 58985?

This sequence also gives the number of occurrences of any digit d > n (thus n < 9) in the list of all numbers from 1 to concatenation(1,...,n) = A007908(n) = A014824(n) = sum_{i=1..n} i*10^(n-i). See A277849, A061217, A277830 etc. - M. F. Hasler, Nov 01 2016, edited Nov 07 2020

Cf. A061217.

a:= n-> (parse(cat((n+1-i)$i=1..n))-parse(cat($1..n)))/9:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 09 2020

Array[(FromDigits@ Apply[Join, Reverse@ #] - FromDigits@ Apply[Join, #])/9 &@ Map[IntegerDigits, Range[#]] &, 19] (* Michael De Vlieger, Nov 12 2020 *)

apply( {A083449(n)=A019566(n)\9}, [1..20]) \\ - M. F. Hasler, Nov 07 2020

For n < 10, a(n) = ceiling((9*n-11)*(10^n+1)/729). - M. F. Hasler, Nov 07 2020

More terms from David Wasserman, Nov 09 2004

A277830 Number of digits '0' in the set of all numbers from 0 to A014824(n) = Sum_{i=1..n} i*10^(n-i) = (0, 1, 12, 123, 1234, 12345, ...).

Original entry on oeis.org

1, 1, 2, 23, 344, 4665, 58986, 713307, 8367628, 96021949, 1083676272, 12071330614, 133058985146, 1454046641578, 15775034317010, 170096022182442, 1824417011947874, 19478738020713306, 207133059219478738, 2194787382318244170, 23182441724417009624, 244170096256515775267

Offset: 0

M. F. Hasler, Nov 01 2016

The first 10 terms are given by a simple explicit formula and linear recurrence, which does not hold for n > 9. Note that A007908 (concat(1..n)) differs from A014824 (a(n) = a(n-1)*10 + n) for n > 9. - M. F. Hasler, Nov 07 2020

M. F. Hasler, Table of n, a(n) for n = 0..100

Cf. A277831 - A277838, A277849, A277635, A272525, A083449, A014824.

print1(c=1);N=0;for(n=1,8,print1(","c+=sum(k=N+1,N=N*10+n,#select(d->d==0,digits(k))))) \\ For purpose of illustration.

apply( A277830(n)={A061217(A014824(n)+!n)+1}, [0..22]) \\ Thanks to Kevin Ryde's formula. - M. F. Hasler, Nov 07 2020

a(n) = A083449(n) + 1 for n <= 9.

a(n) = 1 + A061217(A014824(n)), taking A061217(0)=0. - Kevin Ryde, Nov 07 2020

Incorrect data, b-file, links, formulas and programs deleted by M. F. Hasler, following observations by Kevin Ryde, Nov 07 2020

A155881 a(n) is the number of zeros needed to write the integers 1 through Fibonacci(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 8, 24, 43, 67, 121, 188, 409, 708, 1228, 1946, 4131, 6241, 10525, 17866, 29428, 58369, 87881, 156261, 255242, 412545, 767846, 1280460, 2059307, 3343656, 5510186, 9861418, 16472261, 26422596, 43917688, 73697381, 125281166, 206655249

Offset: 1

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009

Data suggets a(n) ~= 10 ^ (c * n) where c ~= 0.209. - David A. Corneth, Jan 23 2019

			F(9)=34, so writing the numbers F(1)..F(9) requires 3 zeros (one each at 10, 20, and 30), thus a(9)=3.

David A. Corneth, Table of n, a(n) for n = 1..4770

Cf. A000045, A055641, A061217.

A055641 := proc(n) option remember ; local a,d; if n = 0 then RETURN(a) ; fi; a := 0 ; for d in convert(n,base,10) do if d = 0 then a := a+1 ; fi; od: a ; end: A155881 := proc(n) add(A055641(i),i=1..combinat[fibonacci](n)) ; end: for n from 1 do printf("%d,\n",A155881(n)) ; od; # R. J. Mathar, Feb 19 2009

Block[{n = 32, s}, s = DigitCount[Range@ Fibonacci@ n, 10, 0]; Array[Total@ Take[s, Fibonacci@ #] &, n]] (* Michael De Vlieger, Jan 23 2019 *)

nb(n) = #Set(select(x->(x==0), digits(n))); \\ A055641
a(n) = sum(k=1, fibonacci(n), nb(k)); \\ Michel Marcus, Jan 23 2019

a(n) = my(n = fibonacci(n), m=logint(n, 10)); (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2) * sum(j=1, m+1, (n\10^j * (2*n+2 - (1 + n\10^j) * 10 ^ j) - floor(n/10^j+9/10) * (2*n+2 + ((4/5 - floor(n / 10^j + 9 / 10))*10^j)))) \\ David A. Corneth, Jan 23 2019

a(n) = A061217(Fibonacci(n)) = A061217(A000045(n)). - David A. Corneth, Jan 23 2019

8 more terms from R. J. Mathar, Feb 19 2009
9 more terms from Sean A. Irvine, Dec 10 2009
Edited by Jon E. Schoenfield, Jan 22 2019
More terms from David A. Corneth, Jan 23 2019

A364972 Bases >= 2 in which the number of zeros needed to write the numbers 1 through k never equals k for any k.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 27, 30, 32, 35, 37, 38, 39, 40, 41, 43, 45, 48, 49, 53, 54, 57, 58, 59, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 83, 85, 88, 89, 90, 93, 94, 95, 96, 98, 100

Offset: 1

Gregory Marton and Tanya Khovanova, Aug 14 2023

We gave a theoretical upper bound, and experimentally checked values under that bound.

			11 and 13 are not in this sequence because exactly 3152738985031 zeros (expressed here for convenience in decimal) are needed to write the numbers from 1 to 3152738985031, and likewise 3950024143546664 for 13.

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023.
Gregory Marton and Tanya Khovanova, Archive Labeling Sequences: Code (raw for text browsers) for supporting code and tables.

Cf. A061217 (shows 10 is a term).

cp's OEIS Frontend

A196564 Number of odd digits in decimal representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A196563 Number of even digits in decimal representation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A083449 a(n) = A019566(n)/9, where A019566(n) = concat(n,...,1) - concat(1,...,n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A277830 Number of digits '0' in the set of all numbers from 0 to A014824(n) = Sum_{i=1..n} i*10^(n-i) = (0, 1, 12, 123, 1234, 12345, ...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A155881 a(n) is the number of zeros needed to write the integers 1 through Fibonacci(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A364972 Bases >= 2 in which the number of zeros needed to write the numbers 1 through k never equals k for any k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs