A014261 Numbers that contain odd digits only.

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, 111, 113, 115, 117, 119, 131, 133, 135, 137, 139, 151, 153, 155, 157, 159, 171, 173, 175, 177, 179, 191, 193, 195, 197, 199, 311, 313, 315, 317, 319

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy
• look

Comments

Or, numbers whose product of digits is odd.

Complement of A007928; A196563(a(n)) = 0. - Reinhard Zumkeller, Oct 04 2011

If n is represented as a zerofree base-5 number (see A084545) according to n = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j = 0..m} c(d(j))*10^j, where c(k) = 1, 3, 5, 7, 9 for k = 1..5. - Hieronymus Fischer, Jun 06 2012

Examples

			a(10^3) = 13779.
a(10^4) = 397779.
a(10^5) = 11177779.
a(10^6) = 335777779.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for 10-automatic sequences.

Crossrefs

Subsequence of A059708 and of A225985. A066640 and A030096 are subsequences.

Cf. A005408, A010879, A014263, A046034, A084545, A089581, A084984, A001743, A001744, A202267, A202268, A196564.

Cf. A192107, A194181.

Programs

Haskell

a014261 n = a014261_list !! (n-1)
a014261_list = filter (all (`elem` "13579") . show) [1,3..]
-- Reinhard Zumkeller, Jul 05 2011

Magma

[ n : n in [1..129] | IsOdd(&*Intseq(n,10)) ];

Mathematica

Select[Range[400], OddQ[Times@@IntegerDigits[#]] &] (* Alonso del Arte, Feb 21 2014 *)

PARI

is(n)=Set(digits(n)%2)==[1] \\ Charles R Greathouse IV, Jul 06 2017

PARI

a(n)={my(k=1); while(n>5^k, n-=5^k; k++); fromdigits([2*d+1 | d<-digits(5^k+n-1, 5)]) - 3*10^k} \\ Andrew Howroyd, Jan 17 2020

Python

from itertools import islice, count
def A014261(): return filter(lambda n: set(str(n)) <= {'1','3','5','7','9'}, count(1,2))
A014261_list = list(islice(A014261(),20)) # Chai Wah Wu, Nov 22 2021

Python

from itertools import count, islice, product
def agen(): yield from (int("".join(p)) for d in count(1) for p in product("13579", repeat=d))
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 13 2022

Formula

A121759(a(n)) = a(n); A000035(A007959(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2007

From Reinhard Zumkeller, Aug 30 2009: (Start)

a(n+1) - a(n) = A164898(n). - Reinhard Zumkeller, Aug 30 2009

a(n+1) = h(a(n)) with h(x) = 1 + (if x mod 10 < 9 then x + x mod 2 else 10*h(floor(x/10)));

a(n) = f(n, 1) where f(n, x) = if n = 1 then x else f(n-1, h(x)). (End)

From Hieronymus Fischer, Jun 06 2012: (Start)

a(n) = Sum_{j = 0..m-1} ((2*b_j(n)+1) mod 10)*10^j, where b_j(n) = floor((4*n+1-5^m)/(4*5^j)), m = floor(log_5(4*n+1)).

a(1*(5^n-1)/4) = 1*(10^n-1)/9.

a(2*(5^n-1)/4) = 1*(10^n-1)/3.

a(3*(5^n-1)/4) = 5*(10^n-1)/9.

a(4*(5^n-1)/4) = 7*(10^n-1)/9.

a(5*(5^n-1)/4) = 10^n - 1.

a((5^n-1)/4 + 5^(n-1)-1) = (10^n-5)/5.

a(n) = (10^log_5(4*n+1)-1)/9 for n = (5^k-1)/4, k > 0.

a(n) < (10^log_5(4*n+1)-1)/9 for (5^k-1)/4 < n < (5^(k+1)-1)/4, k > 0.

a(n) <= 27/(9*2^log_5(9)-1)*(10^log_5(4*n+1)-1)/9 for n > 0, equality holds for n = 2.

a(n) > 0.776*10^log_5(4*n+1)-1)/9 for n > 0.

a(n) >= A001742(n), equality holds for n = (5^k-1)/4, k > 0.

a(n) = A084545(n) if and only if all digits of A084545(n) are 1, else a(n) > A084545(n).

G.f.: g(x)= (x^(1/4)*(1-x))^(-1) Sum_{j >= 0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 3*z(j) + 5*z(j)^2 + 7*z(j)^3 + 9*z(j)^4)/(1-z(j)^5), where z(j) = x^5^j.

Also: g(x) = (1/(1-x))*(h_(5,0)(x) + 2*h_(5,1)(x) + 2*h_(5,2)(x) + 2*h_(5,3)(x) + 2*h_(5,4)(x) - 9*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j >= 0} 10^j*x^((5^(j+1)-1)/4)*(x^5^j)^k/(1-(x^5^j)^5). (End)

a(n) = A225985(A226091(n)). - Reinhard Zumkeller, May 26 2013

Sum_{n>=1} 1/a(n) = A194181. - Bernard Schott, Jan 13 2022

Extensions

More terms from Robert G. Wilson v, Oct 18 2002

Examples and crossrefs added by Hieronymus Fischer, Jun 06 2012