A135517 a(n) = 2^(A091090(n)-1).

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8

Offset: 0

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

• nonn
• easy

Comments

Also, a(n) = denominator(Euler(n, x) - Euler(n, 1)). - Observation from Peter Luschny, Aug 08 2017, proof from Vladimir Shevelev, Aug 13 2017

Also, a(n) = denominator(Euler(n,x) + Euler(n,0)). - Vladimir Shevelev, Aug 09 2017

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Vladimir Shevelev, Is A290646 = A135517?, Posting to Sequence Fans Mailing List, Aug 13 2017
• Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Crossrefs

This is Guy Steele's sequence GS(2, 5) (see A135416).

Cf. A091090.

Programs

Maple

GS(2,5,200); # see A135416.
a := n -> `if`(n=1 or n mod 2 = 0, 1, 2*a(iquo(n,2))):
seq(a(n), n=0..103); # Peter Luschny, Aug 09 2017

Mathematica

b[n_] := b[n] = Which[n==0, 1, n==1, 1, EvenQ[n], 1, True, b[(n-1)/2] + 1]; a[n_] := 2^(b[n+1]-1); Array[a,103,0] (* Jean-François Alcover, Aug 12 2017 *)

PARI

a(n)=my(m=valuation(n+1,2)); 2^if(n>>m, m, m-1) \\ Charles R Greathouse IV, Aug 15 2017

Python

def A135517(n): return (1<<(~(n+1)&n).bit_length()-(not n&(n+1))) if n else 1 # Chai Wah Wu, Sep 18 2024

Formula

For n >= 1, a(n) = 2^max_{odd k=1..n} (A007814(k+1) - t(n,k) - delta(n,k)), where delta(n,k) is the Kronecker symbol: delta(i,j) is 1 if i=j and 0 otherwise, and t(n,k) is the number of carries which appear in the addition of k and n-k in base 2. This allows us to answer in the affirmative the author's question (for a proof see Shevelev's link and its continuations). - Vladimir Shevelev, Aug 15 2017

Extensions

Entry revised by N. J. A. Sloane, Aug 31 2017