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.

A103391 "Even" fractal sequence for the natural numbers: Deleting every even-indexed term results in the same sequence.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42, 22, 43, 12, 44, 23, 45, 7, 46, 24, 47, 13, 48, 25, 49, 3, 50, 26, 51, 14, 52, 27, 53, 8
Offset: 1

Views

Author

Eric Rowland, Mar 20 2005

Keywords

Comments

A003602 is the "odd" fractal sequence for the natural numbers.
Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(A005940(i)) = A348717(A005940(j)) for all i, j >= 1. A365718 is an analogous sequence related to A356867 (Doudna variant D(3)). - Antti Karttunen, Sep 17 2023

Links

Crossrefs

Cf. A003602, A005940, A025480, A220466, A286387, A353368 (Dirichlet inverse).
Cf. also A110962, A110963, A365718.
Differs from A331743(n-1) for the first time at n=192, where a(192) = 97, while A331743(191) = 23.
Differs from A351460.

Programs

  • Haskell
    -- import Data.List (transpose)
a103391 n = a103391_list !! (n-1)
a103391_list = 1 : ks where
   ks = concat $ transpose [[2..], ks]
-- Reinhard Zumkeller, May 23 2013
  • Maple
    nmax := 82: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to ceil(nmax/(p+2))+1 do a((2*n-3)*2^p+1) := n od: od: a(1) := 1: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 28 2013
  • Mathematica
    a[n_] := ((n-1)/2^IntegerExponent[n-1, 2] + 3)/2; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)
  • PARI
    A003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1))); \\ Antti Karttunen, Feb 05 2020
  • Python
    def v(n): b = bin(n); return len(b) - len(b.rstrip("0"))
def b(n): return (n//2**v(n)+1)//2
def a(n): return 1 if n == 1 else 1 + b(n-1)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, May 29 2022
  • Python
    def A103391(n): return (n-1>>(n-1&-n+1).bit_length())+2 if n>1 else 1 # Chai Wah Wu, Jan 04 2024

Formula

For n > 1, a(n) = A003602(n-1) + 1. - Benoit Cloitre, May 26 2007, indexing corrected by Antti Karttunen, Feb 05 2020
a((2*n-3)*2^p+1) = n, p >= 0 and n >= 2, with a(1) = 1. - Johannes W. Meijer, Jan 28 2013
Sum_{k=1..n} a(k) ~ n^2/6. - Amiram Eldar, Sep 24 2023

Extensions

Data section extended up to a(105) (to better differentiate from several nearby sequences) by Antti Karttunen, Feb 05 2020

A353366 Dirichlet inverse of A110963, which is a fractalization of Kimberling's paraphrases sequence (A003602).

Original entry on oeis.org

1, -1, -1, 0, -2, 1, -1, 0, -2, 2, -2, 0, -4, 1, 3, 0, -5, 2, -3, 0, -4, 2, -2, 0, -3, 4, 1, 0, -8, -3, -1, 0, -5, 5, -1, 0, -10, 3, 5, 0, -11, 4, -6, 0, -4, 2, -2, 0, -12, 3, 3, 0, -14, -1, 4, 0, -9, 8, -8, 0, -16, 1, 14, 0, -1, 5, -9, 0, -14, 1, -5, 0, -19, 10, -4, 0, -16, -5, -3, 0, -12, 11, -11, 0, -2, 6, 10
Offset: 1

Views

Author

Antti Karttunen, Apr 18 2022

Keywords

Links

Crossrefs

Cf. A003602, A110963, A353367.
Cf. also A349134, A353368.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];
  • Python
    from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A353366(n): return 1 if n==1 else -sum(((1+(m:=d>>(~d&d-1).bit_length())>>(m+1&-m-1).bit_length())+1)*A353366(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 04 2024

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A110963(n/d) * a(d).
a(n) = A353367(n) - A110963(n).

A353369 Sum of A103391 ("even fractal sequence") and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 4, 4, 8, 0, 4, 0, 12, 8, 9, 0, 0, 0, 12, 12, 16, 0, 16, 4, 12, 0, 14, 0, 12, 0, 16, 16, 8, 12, 36, 0, 24, 12, 20, 0, 0, 0, 24, 4, 28, 0, 24, 9, 12, 8, 38, 0, 56, 16, 30, 24, 20, 0, 34, 0, 36, -8, 32, 12, -8, 0, 60, 28, 36, 0, 20, 0, 24, 8, 44, 24, 52, 0, 44, 28, 16, 0, 74, 8, 48, 20, 44, 0, 52
Offset: 1

Views

Author

Antti Karttunen, Apr 18 2022

Keywords

Links

Crossrefs

Cf. A003602, A103391, A353368.
Cf. also A349135, A353367.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1)));
v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
A353368(n) = v353368[n];
A353369(n) = (A103391(n)+A353368(n));

Formula

a(n) = A103391(n) + A353368(n).
For n > 1, a(n) = -Sum_{d|n, 1A103391(d) * A353368(n/d).
Showing 1-3 of 3 results.

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