login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A129527 a(2n) = a(n) + 2n, a(2n+1) = 2n + 1. 16
0, 1, 3, 3, 7, 5, 9, 7, 15, 9, 15, 11, 21, 13, 21, 15, 31, 17, 27, 19, 35, 21, 33, 23, 45, 25, 39, 27, 49, 29, 45, 31, 63, 33, 51, 35, 63, 37, 57, 39, 75, 41, 63, 43, 77, 45, 69, 47, 93, 49, 75, 51, 91, 53, 81, 55, 105, 57, 87, 59, 105, 61, 93, 63, 127, 65, 99, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sum of odd part of n and its double, fourfold, eightfold etc. <= n.

Starting with 1 = the ruler function triangle A115361 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009

Starting with 1 and parsed into subsets of 1, 2, 4, 8, ... terms, sum of terms in the subsets = A006516: (1, 6, 28, 120, ...). Example: 120 = (15 + 9 + 15 + 11 + 21 + 13 + 21 + 15). - Gary W. Adamson Mar 18 2011

a(n) = Sum(even divisors of 2(n-1) not including 2(n-1) that are obtained dividing repeatedly by 2) + (greatest odd divisor of 2(n-1), including 1), for the initial case 2(1-1)=0 will be set to 0). E.g., (offset is starting with n=1) a(3) = Sum(even divisors of 2*(3-1)=2*2=4 not including 4 obtained dividing repeatedly by 2) + greatest odd divisor of 4 = (2)+(1)=3; a(4) = Sum(even divisors of 2*(4-1)=6 not including 6 obtained dividing repeatedly by 2) + greatest odd divisor of 6 = (0) + (3) = 3; a(5) = Sum(even divisors of 2*(5-1)=8 not including 8 obtained dividing repeatedly by 2) + greatest odd divisor of 8 = (4+2) + (1) = 7, etc. - David Morales Marciel, Dec 21 2015

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: Sum_{k>=0} x^(2^k)/(1-x^(2^k))^2.

Dirichlet g.f.: zeta(s-1)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007

G.f. satisfies g(x) = g(x^2) + x/(1-x)^2. - Robert Israel, Dec 20 2015

n <= a(n) <= 2n - 1 for n > 0. - Charles R Greathouse IV, Feb 09 2016

Conjecture: a(n) = 2*n-A000265(n) for n > 0. - Velin Yanev, Jun 23 2017. [Joerg Arndt, Jun 23 2017: For odd n the conjecture holds, for even n induction should work.

Andrey Zabolotskiy, Aug 03 2017: Confirm: induction works, the conjecture holds for all n.]

a(n) for n > 0 is multiplicative with a(2^e) = 2^(e+1)-1 and a(p^e) = p^e for prime p > 2 and e >= 0. - Werner Schulte, Jul 02 2018

Inverse Moebius transform of A062570. - Andrew Howroyd, Aug 07 2018

Sum_{k=1..n} a(k) ~ 2*n^2/3. - Vaclav Kotesovec, Jun 11 2020

MAPLE

f:= proc(n) option remember;

  if n::odd then n else n + procname(n/2) fi

end proc:

f(0):= 0:

seq(f(n), n=0..100); # Robert Israel, Dec 20 2015

MATHEMATICA

a[n_] := a[n] = If[EvenQ@ n, a[n/2] + n, n]; {0}~Join~Array[a, 67] (* Michael De Vlieger, Jun 26 2017 *)

PROG

(PARI) a(n)=sum(k=0, valuation(n, 2), n/2^k)

(PARI) a(n)=if(n<2, return(n)); my(k=valuation(n, 2)); 2*n-n>>k \\ Charles R Greathouse IV, Feb 09 2016

(PARI) a(n)=sumdiv(n, d, eulerphi(2*d)) \\ Andrew Howroyd, Aug 07 2018

CROSSREFS

Row sums of A129265 and A129559.

Cf. A000265, A006516, A062570, A115361.

Sequence in context: A161427 A098688 A129266 * A245550 A318461 A085379

Adjacent sequences:  A129524 A129525 A129526 * A129528 A129529 A129530

KEYWORD

nonn,easy,mult

AUTHOR

Ralf Stephan, May 29 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 23:44 EST 2020. Contains 338755 sequences. (Running on oeis4.)