A129527 a(2n) = a(n) + 2n, a(2n+1) = 2n + 1.

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

Offset: 0

Author

Ralf Stephan, May 29 2007

Keywords

• nonn
• easy
• mult

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

For n >=1, a(n) is the sum of divisors d of n such that n/d is a power of 2. - Amiram Eldar, Nov 17 2022

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Cristina Ballantine and Mircea Merca, Plane Partitions and Divisors, Symmetry (2024), Vol. 16, Iss. 5. See page 3.
• Mircea Merca and Emil Simion, n-Color Partitions into Distinct Parts as Sums over Partitions, Symmetry (2023) Vol. 15, Iss. 11.

Crossrefs

Row sums of A129265 and A129559.

Cf. A000265, A006516, A062570, A115361.

Programs

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 *)

PARI

a(n)=if (n==0, 0, sum(k=0,valuation(n,2),n/2^k)); \\ corrected by Michel Marcus, Dec 22 2021

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

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

a(n) = A038712(n)*A000265(n), for n > 0. - Ivan N. Ianakiev, Feb 24 2025