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.

Showing 1-3 of 3 results.

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

A192370 Sum of all the n-digit numbers whose digits are all even and nonzero: 2,4,6,8.

Original entry on oeis.org

20, 880, 35520, 1422080, 56888320, 2275553280, 91022213120, 3640888852480, 145635555409920, 5825422221639680, 233016888886558720, 9320675555546234880, 372827022222184939520, 14913080888888739758080, 596523235555554959032320, 23860929422222219836129280
Offset: 1

Views

Author

Bernard Schott, Dec 31 2012

Keywords

Comments

A192107 is the similar sequence when all the digits are odd.
A220094 is the similar sequence with the digits belonging to {1, 2, 3, 4, 5, 6, 7, 8, 9}.

Examples

			a(1) = 2 + 4 + 6 + 8 = 20.
a(2) = 22 + 24 + 26 + 28 + 42 + ... + 68 + 82 + 84 + 86 + 88 = 880.

Links

Crossrefs

Cf. A192107, A220094, A045926.

Programs

  • Maple
    A:=seq(5 * 4^n *(10^n-1)/9,n=1..20);
  • Mathematica
    Table[(5*4^n*(10^n - 1))/9, {n, 20}] (* T. D. Noe, Dec 31 2012 *)
  • PARI
    a(n)=(5*4^n*(10^n-1))/9 \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = (5 * 4^n * (10^n-1))/9.
From Colin Barker, Jan 04 2013: (Start)
a(n) = 44*a(n-1) - 160*a(n-2).
G.f.: 20*x/((4*x-1)*(40*x-1)). (End)

A220094 Sum of the n-digit base-ten numbers whose digits are nonzero.

Original entry on oeis.org

45, 4455, 404595, 36446355, 3280467195, 295244704755, 26572047342795, 2391484476085155, 215233604784766395, 19371024448062897555, 1743392200482566077995, 156905298044843094701955, 14121476824048587852317595, 1270932914164487290670858355
Offset: 1

Views

Author

Bernard Schott, Dec 04 2012

Keywords

Comments

For n >= 1, a(n) is the sum of the numbers with n digits in base ten whose digits belong to the set {1,2,3,4,5,6,7,8,9}.
If E_n is the set of the numbers with n digits in base ten whose digits belong to {1,2,3,4,5,6,7,8,9}, then card(E_n) = 9^n (see A001019).

Examples

			For n=2, in base ten, a(2) = 11+12+...+19+21+...+89+91+...+98+99 = 4455.

References

  • A. Ducos, ElĂ©ments fondamentaux de Math Sup, Ellipses, 1994, exercice 9, p. 126.

Links

Crossrefs

Cf. A192107, A192370.

Programs

  • Maple
    := n->5*9^(n-1)*(10^n-1);
  • Mathematica
    Table[5*9^(n - 1)*(10^n - 1), {n, 20}] (* T. D. Noe, Dec 31 2012 *)
  • PARI
    a(n)=5*9^(n-1)*(10^n-1) \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = 5*9^(n-1)*(10^n-1).
Generalization to base b with n-digit numbers whose digits belong to {1,2,...,b-1}: a_b(n) = (b/2)*(b-1)^(n-1)*(b^n-1).
From Colin Barker, Jan 04 2013: (Start)
a(n) = 99*a(n-1) - 810*a(n-2).
G.f.: 45*x/((9*x-1)*(90*x-1)). (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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