A037227 If n = 2^m*k, k odd, then a(n) = 2*m+1.

1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 13, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 7, 1, 3, 1, 5, 1, 3, 1, 11, 1, 3, 1, 5, 1, 3

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

Comments

Take the number of rightmost zeros in the binary expansion of n, double it, and increment it by 1. - Ralf Stephan, Aug 22 2013

Gives the maximum possible number of n X n complex Hermitian matrices with the property that all of their nonzero real linear combinations are nonsingular (see Adams et al. reference). - Nathaniel Johnston, Dec 11 2013

Links

• T. D. Noe, Table of n, a(n) for n=1..1024
• J. F. Adams, P. D. Lax, and R. S. Phillips, On matrices whose real linear combinations are nonsingular, Proceedings of the American Mathematical Society, 16:318-322, 1965.
• D. B. Shapiro, Problem 10456: Anticommuting Matrices, Amer. Math. Monthly, 105 (1998), 565-566.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A001511, A002487, A007814, A005408, A016825, A017113, A075300, A051062, A220466.

Programs

Haskell

a037227 = (+ 1) . (* 2) . a007814  -- Reinhard Zumkeller, Jun 30 2012

Magma

[2*Valuation(n, 2)+1: n in [1..120]]; // Vincenzo Librandi, Jun 19 2019

Maple

nmax:=102: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p):= 2*p+1: od: od: seq(a(n), n=1..nmax);  # Johannes W. Meijer, Feb 07 2013

Mathematica

a[n_] := Sum[(-1)^(d+1)*MoebiusMu[d]*DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Dec 31 2012, after Vladeta Jovovic *)
f[n_]:=Module[{z=Last[Split[IntegerDigits[n,2]]]},If[Union[z]={0},2* Length[ z]+1,1]]; Array[f,110] (* Harvey P. Dale, Jun 16 2019, after Ralf Stephan *)
Table[2 IntegerExponent[n, 2] + 1, {n, 120}] (* Vincenzo Librandi, Jun 19 2019 *)

PARI

a(n)=2*valuation(n,2)+1 \\ Charles R Greathouse IV, May 21 2015

Python

def A037227(n): return ((~n & n-1).bit_length()<<1)+1 # Chai Wah Wu, Jul 05 2022

R

maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow){
for(k in 0:(2^m-1)) {
   a[2^(m+1)    +k] <- a[2^m+k]
   a[2^(m+1)+2^m+k] <- a[2^m+k]
}
   a[2^(m+1)      ] <- a[2^(m+1)] + 2
}
a
# Yosu Yurramendi, May 21 2015

Formula

a(n) = Sum_{d divides n} (-1)^(d+1)*mu(d)*tau(n/d). Multiplicative with a(p^e) = 2*e+1 if p = 2; 1 if p > 2. - Vladeta Jovovic, Apr 27 2003

a(n) = a(n-1)+(-1)^n*(a(floor(n/2))+1). - Vladeta Jovovic, Apr 27 2003

a(2*n) = a(n) + 2, a(2*n+1) = 1. a(n) = 2*A007814(n) + 1. - Ralf Stephan, Oct 07 2003

a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7. - Reinhard Zumkeller, Jun 30 2012

a((2*n-1)*2^p) = 2*p+1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 07 2013

From Peter Bala, Feb 07 2016: (Start)

a(n) = ( A002487(n-1) + A002487(n+1) )/A002487(n).

a(n*2^(k+1) + 2^k) = 2*k + 1 for n,k >= 0; thus a(2*n+1) = 1, a(4*n+2) = 3, a(8*n+4) = 5, a(16*n+8) = 7 and so on. Note the square array ( n*2^(k+1) + 2^k - 1 )n, k>=0 is the transpose of A075300.

G.f.: Sum_{n >= 0} (2*n + 1)*x^(2^n)/(1 - x^(2^(n+1))). (End)

a(n) = 2*floor(A002487(n-1)/A002487(n))+1 for n > 1. - I. V. Serov, Jun 15 2017

From Amiram Eldar, Nov 29 2022: (Start)

Dirichlet g.f.: zeta(s)*(2^s+1)/(2^s-1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3. (End)

Extensions

More terms from Erich Friedman