cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A014261 Numbers that contain odd digits only.

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, 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

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Author

N. J. A. Sloane

Keywords

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

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

A225985 List the positive numbers, remove even digits (including zeros) from each term; sequence = remaining terms.

Original entry on oeis.org

1, 3, 5, 7, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 3, 5, 7, 9, 3, 31, 3, 33, 3, 35, 3, 37, 3, 39, 1, 3, 5, 7, 9, 5, 51, 5, 53, 5, 55, 5, 57, 5, 59, 1, 3, 5, 7, 9, 7, 71, 7, 73, 7, 75, 7, 77, 7, 79, 1, 3, 5, 7, 9, 9, 91, 9, 93, 9, 95, 9, 97, 9, 99, 1, 11, 1, 13
Offset: 1

Views

Author

Dave Durgin, May 22 2013

Keywords

Examples

			The natural numbers with at least one odd digit in their decimal representation are: 1, 3, 5, 7, 9, 10, 11, 12, 13, ...
By excluding their even digits, we obtain: 1, 3, 5, 7, 9, 1, 11, 1, 13, ...
Hence: a(1)=1, a(2)=3, a(3)=5, a(4)=7, a(5)=9, a(6)=1, a(7)=11, a(8)=1, a(9)=13, .... [Example corrected by _Paul Tek_, May 24 2013]

Links

Crossrefs

Cf. A038573.
Cf. A014261 (duplicates removed), A226091.

Programs

  • Haskell
    a225985 n = a225985_list !! (n-1)
a225985_list = map read $ filter (not . null) $
    map (filter (`elem` "13579") . show) [0..] :: [Integer]
-- Reinhard Zumkeller, May 26 2013
  • Mathematica
    FromDigits[DeleteCases[IntegerDigits[#],?EvenQ]]&/@Range[200]/. (0-> Nothing) (* _Harvey P. Dale, Apr 04 2017 *)
  • Python
    from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        removed  = "".join(d if d in "13579" else "" for d in str(n))
        if removed != "": yield int(removed)
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 05 2022

Formula

a(A226091(n)) = A014261(n). - Reinhard Zumkeller, May 26 2013

Extensions

Definition clarified by N. J. A. Sloane, Aug 06 2022 at the suggestion of Michel Marcus
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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