A196564 -id:A196564 - cp's OEIS frontend

A104638 Number of odd digits in n-th prime.

0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Zak Seidov, Mar 18 2005

The only zero is the first term. Sequence is unbounded. - Zak Seidov, Jan 12 2016
From Robert Israel, Jan 12 2016: (Start)
For any N, the asymptotic density of terms >= N is 1.
On the other hand, a(n) = 2 if prime(n) is in A159352, which is conjectured to be infinite.
Record values: a(2) = 1, a(5) = 2, a(30) = 3, a(187) = 4, a(1346) = 5, a(10545) = 6, a(86538) = 7, a(733410) = 8.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A104637, A159352.

Cf. A154764 (1 odd digit), A155071 (2 odd digits), A030096 (all digits odd).

seq(nops(select(type, convert(ithprime(i),base,10),odd)),i=1..100); # Robert Israel, Jan 12 2016
# alternative
A104638 := proc(n)
    local a,dgs,d ;
    ithprime(n) ;
    dgs := convert(%,base,10) ;
    a := 0 ;
    for d in dgs do
        a := a+modp(d,2) ;
    end do:
    a ;
end proc:
seq(A104638(n),n=1..40) ; # R. J. Mathar, Jul 13 2025

Table[Count[IntegerDigits[Prime[n]],?OddQ],{n,100}] (* _Harvey P. Dale, Jan 22 2012 *)
Table[Total[Mod[IntegerDigits[Prime[n]], 2]], {n, 100}] (* Vincenzo Librandi, Jan 13 2016 *)

a(n)=vecsum(digits(prime(n)%2)) \\ Zak Seidov, Jan 12 2016

a(n) = A196564(A000040(n)). - Michel Marcus, Oct 05 2013

A104640 Number of odd digits in n^3.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Mar 18 2005

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

Cf. A104639, A196564.

Table[Count[IntegerDigits[n^3],?OddQ],{n,100}] (* _Harvey P. Dale, Jun 23 2017 *)

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

a(n) = A196564(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

A338741 When a(n) is odd, a(n) is the number of odd digits present so far in the sequence, a(n) included.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 3, 12, 5, 14, 7, 16, 9, 11, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 32, 21, 34, 23, 36, 25, 38, 27, 40, 42, 44, 46, 48, 50, 29, 31, 33, 35, 37, 39, 52, 41, 54, 43, 56, 45, 58, 47, 60, 62, 64, 66, 68, 70, 49, 51, 53, 55, 57, 59, 72, 61, 74, 63, 76, 65, 78, 67, 80, 82, 84, 86, 88, 90, 69

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 (19 for instance).

			The first odd term is a(2) = 1 and there is indeed 1 odd digit so far in the sequence (1 itself);
The next odd term is a(8) = 3 and there are now 3 odd digits so far (1, 1 and 3);
The next odd term is a(10) = 5 and there are now 5 odd digits so far (1, 1, 3, 1 and 5);
...
The next odd term is a(18) = 17 and there are indeed 17 odd digits so far in the sequence (1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 1, 1, 3, 1, 5, 1, 7); etc.

Cf. A338742, A338743, A338744, A338745, A338746 (variants on the same idea), A196564.

Block[{a = {0}, c = 0}, Do[Block[{k = 1, s}, While[If[OddQ[k], Nand[FreeQ[a, k], k == c + Set[s, Total@ DigitCount[k, 10, {1, 3, 5, 7, 9}]]], ! FreeQ[a, k]], k++]; If[OddQ[k], c += s, c += Total@ DigitCount[k, 10, {1, 3, 5, 7, 9}]]; AppendTo[a, k]], {i, 79}]; a] (* Michael De Vlieger, Nov 06 2020 *)

A338744 When a(n) is even, a(n) is the number of odd digits present so far in the sequence, a(n) included.

Original entry on oeis.org

0, 1, 3, 2, 5, 7, 4, 9, 11, 13, 10, 15, 17, 19, 21, 18, 23, 25, 20, 27, 29, 22, 31, 24, 33, 26, 35, 28, 37, 39, 41, 34, 43, 36, 45, 38, 47, 49, 40, 51, 42, 53, 44, 55, 46, 57, 48, 59, 61, 52, 63, 54, 65, 56, 67, 58, 69, 71, 73, 75, 77, 79, 70, 81, 72, 83, 74, 85, 76, 87, 78, 89, 91, 93, 95, 97, 99, 90, 101

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 (6 for instance).

			The first even term is a(1) = 0 and there is indeed 0 odd digit so far in the sequence;
The next even term is a(4) = 2 and there are now 2 odd digits so far (1 and 3);
The next even term is a(7) = 4 and there are now 4 odd digits so far (1, 3, 5 and 7);
...
The even term a(11) = 10 and there are indeed 10 odd digits in the sequence so far (1, 3, 5, 7, 9, 1, 1, 1, 3 and 1); etc.

Cf. A338741, A338742, A338743, A338745, A338746 (variants on the same idea), A196564.

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, {1, 3, 5, 7, 9}]]], ! FreeQ[a, k]], k++]; If[EvenQ[k], c += s, c += Total@ DigitCount[k, 10, {1, 3, 5, 7, 9}]]; AppendTo[a, k]], {i, 78}]; 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

A007957 Numbers that contain an odd digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

R. Muller

Complement of A014263; A196564(a(n)) > 0; A103181(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.

Cf. A014263, A103181, A196564.

import Data.List (findIndices)
a007957 n = a007957_list !! (n-1)
a007957_list = findIndices (> 0) a196564_list
a196564 n = length [d | d <- show n, d `elem` "13579"]
a196564_list = map a196564 [0..]
-- Reinhard Zumkeller, Oct 04 2011

Select[Range[100],Count[IntegerDigits[#],?OddQ]>0&] (* _Harvey P. Dale, Sep 06 2017 *)

is(n)=my(v=vecsort(eval(Vec(Str(n)))%2,,8));v[#v] \\ Charles R Greathouse IV, Jul 25 2012

a(n) = n + O(n^0.69897...) where the constant is A153268. - Charles R Greathouse IV, Jul 25 2012

A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20

Offset: 0

N. J. A. Sloane, Feb 03 2005

This is the total number of digits = 9 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

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

Partial sums of A102683.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
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]>=9 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..105); # Emeric Deutsch, Feb 23 2005

Accumulate[DigitCount[Range[0,100],10,9]] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = sum(k=0, n, #select(x->(x==9), digits(k))); \\ Michel Marcus, Oct 03 2023

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/10)*(2n + 2 - (4/5 + floor(n/10^j + 1/10))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102683(n) + (1/2)*Sum_{j=1..m+1} ((-4/5*floor(n/10^j + 1/10) + floor(n/10^j))*10^j - (floor(n/10^j + 1/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = m*10^(m-1).
(this is total number of digits = 9 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(9*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

Definition revised by N. J. A. Sloane, Mar 30 2018

cp's OEIS Frontend

A104638 Number of odd digits in n-th prime.

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A104640 Number of odd 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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A338741 When a(n) is odd, a(n) is the number of odd digits present so far in the sequence, a(n) included.

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A338744 When a(n) is even, a(n) is the number of odd 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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A352547 Numbers having more odd than even digits when written in base 10.

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