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.

A032924 Numbers whose ternary expansion contains no 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 13, 14, 16, 17, 22, 23, 25, 26, 40, 41, 43, 44, 49, 50, 52, 53, 67, 68, 70, 71, 76, 77, 79, 80, 121, 122, 124, 125, 130, 131, 133, 134, 148, 149, 151, 152, 157, 158, 160, 161, 202, 203, 205, 206, 211, 212, 214, 215, 229, 230, 232, 233, 238, 239
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Complement of A081605. - Reinhard Zumkeller, Mar 23 2003
Subsequence of A154314. - Reinhard Zumkeller, Jan 07 2009
The first 28 terms are the range of A059852 (Morse codes for letters, when written in base 3) union {44, 50} (which correspond to Morse codes of Ü and Ä). Subsequent terms represent the Morse code of other symbols in the same coding. - M. F. Hasler, Jun 22 2020

Links

Crossrefs

Cf. A005823, A005836, A065361, A007089, A081608, A132140, A132141.
Zeroless numbers in some other bases <= 10: A000042 (base 2), A023705 (base 4), A248910 (base 6), A255805 (base 8), A255808 (base 9), A052382 (base 10).

Programs

  • Haskell
    a032924 n = a032924_list !! (n-1)
a032924_list = iterate f 1 where
   f x = 1 + if r < 2 then x else 3 * f x'  where (x', r) = divMod x 3
-- Reinhard Zumkeller, Mar 07 2015, May 04 2012
  • Maple
    f:= proc(n) local L,i,m;
   L:= convert(n,base,2);
   m:= nops(L);
   add((1+L[i])*3^(i-1),i=1..m-1);
end proc:
map(f, [$2..101]); # Robert Israel, Aug 04 2015
  • Mathematica
    Select[Range@ 240, Last@ DigitCount[#, 3] == 0 &] (* Michael De Vlieger, Aug 05 2015 *)
Flatten[Table[FromDigits[#,3]&/@Tuples[{1,2},n],{n,5}]] (* Harvey P. Dale, May 28 2016 *)
  • PARI
    apply( {A032924(n)=if(n<3,n,3*self()((n-1)\2)+2-n%2)}, [1..99]) \\ M. F. Hasler, Jun 22 2020
  • PARI
    a(n) = fromdigits(apply(d->d+1,binary(n+1)[^1]), 3); \\ Kevin Ryde, Jun 23 2020
  • Python
    def a(n): return sum(3**i*(int(b)+1) for i, b in enumerate(bin(n+1)[:2:-1]))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Aug 15 2022
  • Python
    def is_A032924(n):
    while n > 2:
       n,r = divmod(n,3)
       if r==0: return False
    return n > 0
print([n for n in range(250) if is_A032924(n)]) # M. F. Hasler, Feb 15 2023
  • Python
    def A032924(n): return int(bin(m:=n+1)[3:],3) + (3**(m.bit_length()-1)-1>>1) # Chai Wah Wu, Oct 13 2023

Formula

a(n) = A107680(n) + A107681(n). - Reinhard Zumkeller, May 20 2005
A081604(A107681(n)) <= A081604(A107680(n)) = A081604(a(n)) = A000523(n+1). - Reinhard Zumkeller, May 20 2005
A077267(a(n)) = 0. - Reinhard Zumkeller, Mar 02 2008
a(1)=1, a(n+1) = f(a(n)+1,a(n)+1) where f(x,y) = if x<3 and x<>0 then y, else if x mod 3 = 0 then f(y+1,y+1), else f(floor(x/3),y). - Reinhard Zumkeller, Mar 02 2008
a(2*n) = a(2*n-1)+1, n>0. - Zak Seidov, Jul 27 2009
A212193(a(n)) = 0. - Reinhard Zumkeller, May 04 2012
a(2*n+1) = 3*a(n)+1. - Robert Israel, Aug 05 2015
G.f.: x/(1-x)^2 + Sum_{m >= 1} 3^(m-1)*x^(2^(m+1)-1)/((1-x^(2^m))*(1-x)). - Robert Israel, Aug 04 2015
A065361(a(n)) = n. - Rémy Sigrist, Feb 06 2023
Sum_{n>=1} 1/a(n) = 3.4977362637842652509313189236131190039368413460747606236619907531632476445332666030262441154353753276457... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A062050 n-th chunk consists of the numbers 1, ..., 2^n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
Offset: 1

Views

Author

Marc LeBrun, Jun 30 2001

Keywords

Comments

a(k) is the distance between k and the largest power of 2 not exceeding k, where k = n + 1. [Consider the sequence of even numbers <= k; after sending the first term to the last position delete all odd-indexed terms; the final term that remains after iterating the process is the a(k)-th even number.] - Lekraj Beedassy, May 26 2005
Triangle read by rows in which row n lists the first 2^(n-1) positive integers, n >= 1; see the example. - Omar E. Pol, Sep 10 2013

Examples

			From _Omar E. Pol_, Aug 31 2013: (Start)
Written as irregular triangle with row lengths A000079:
  1;
  1, 2;
  1, 2, 3, 4;
  1, 2, 3, 4, 5, 6, 7, 8;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
  ...
Row sums give A007582.
(End)

Links

Crossrefs

Cf. A053644, A053645.
Cf. A092754.

Programs

  • Haskell
    a062050 n = if n < 2 then n else 2 * a062050 n' + m - 1
            where (n',m) = divMod n 2
-- Reinhard Zumkeller, May 07 2012
  • Maple
    A062050 := proc(n) option remember; if n < 4 then return [1, 1, 2][n] fi;
2*A062050(floor(n/2)) + irem(n,2) - 1 end:
seq(A062050(n), n=1..89); # Peter Luschny, Apr 27 2020
  • Mathematica
    Flatten[Table[Range[2^n],{n,0,6}]] (* Harvey P. Dale, Oct 12 2015 *)
  • PARI
    a(n)=floor(n+1-2^logint(n,2))
  • PARI
    a(n)= n - 1<Ruud H.G. van Tol, Dec 13 2024
  • Python
    def A062050(n): return n-(1<Chai Wah Wu, Jan 22 2023

Formula

a(n) = A053645(n) + 1.
a(n) = n - msb(n) + 1 (where msb(n) = A053644(n)).
a(n) = 1 + n - 2^floor(log(n)/log(2)). - Benoit Cloitre, Feb 06 2003; corrected by Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 25 2008
G.f.: 1/(1-x) * ((1-x+x^2)/(1-x) - Sum_{k>=1} 2^(k-1)*x^(2^k)). - Ralf Stephan, Apr 18 2003
a(1) = 1, a(2*n) = 2*a(n) - 1, a(2*n+1) = 2*a(n). - Ralf Stephan, Oct 06 2003
A005836(a(n+1)) = A107681(n). - Reinhard Zumkeller, May 20 2005
a(n) = if n < 2 then n else 2*a(floor(n/2)) - 1 + n mod 2. - Reinhard Zumkeller, May 07 2012
Without the constant 1, Ralf Stephan's g.f. becomes A(x) = x/(1-x)^2 - (1/(1-x)) * Sum_{k>=1} 2^(k-1)*x^(2^k) and satisfies the functional equation A(x) - 2*(1+x)*A(x^2) = x*(1 - x - x^2)/(1 - x^2). - Petros Hadjicostas, Apr 27 2020
For n > 0: a(n) = (A006257(n) + 1) / 2. - Frank Hollstein, Oct 25 2021

A107680 Repeating k-th ternary repunit (A003462) 2^k times, k >= 0.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 4, 13, 13, 13, 13, 13, 13, 13, 13, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121
Offset: 0

Views

Author

Reinhard Zumkeller, May 20 2005

Keywords

Comments

a(n) is the greatest ternary repunit that is not greater than the n-th number with no 2 in ternary representation.

Examples

			k=1: A003462(1) = (3^1-1)/2 = 1, therefore a(1) = a(2^1) = 1;
k=2: A003462(2) = (3^2-1)/2 = 4, therefore a(2+1) = a(2+2) =
a(2+3) = a(2+2^2) = 4.

Crossrefs

Cf. A007089, A003462 (repunits in base 3), A000523 (number of digits in binary representation of n).

Programs

  • Mathematica
    With[{nn=5},Flatten[Table[#[[1]],{#[[2]]}]&/@Thread[{Table[FromDigits[ PadRight[{},n,1],3],{n,nn}],2^Range[nn]}]]] (* Harvey P. Dale, Jan 04 2013 *)
  • PARI
    apply( {A107680(n)=3^exponent(n+1)\2}, [0..66]) \\ M. F. Hasler, Jun 22 2020
  • Python
    def A107680(n): return 3**((n+1).bit_length()-1)-1>>1 # Chai Wah Wu, Nov 07 2024

Formula

A032924(n) = a(n) + A107681(n);
A081604(A107681(n)) <= A081604(a(n)) = A081604(A032924(n)) = A000523(n+1).
a(n) = A003462(A000523(n+1)).

Extensions

Corrected by T. D. Noe, Oct 25 2006
Extended to a(0) = 0 by M. F. Hasler, Jun 23 2020
Showing 1-3 of 3 results.

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

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