A129265 -id:A129265 - cp's OEIS frontend

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

Ralf Stephan, May 29 2007

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

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.

Row sums of A129265 and A129559.

Cf. A000265, A006516, A062570, A115361.

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

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

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

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

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

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

A129264 Triangle read by rows: A000012 * A115361 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Apr 06 2007

Triangle read by rows, A000012 * A115361; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1; ...] and A115361 = the ruler function triangle.
The operation A000012 * A115361 takes partial sums of A115361 column terms starting from the top. [Gary W. Adamson, Nov 27 2009]
Eigensequence of the triangle = A089067: (1, 3, 5, 13, 23, 51, 97, 207, ...). [Gary W. Adamson, Nov 27 2009]
Row sums = A005187: (1, 3, 4, 7, 8, 10, 11, 15, ...).
Left column = A070939 starting (1, 2, 2, 3, 3, 3, 3, 4, 4, 4, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  3, 2, 1, 1, 1;
  3, 2, 2, 1, 1, 1;
  3, 2, 2, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1;
  4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A005187, A070939, A089067, A115361, A129265.

Incorrect a(79) removed by Georg Fischer, Jul 04 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A129264 Triangle read by rows: A000012 * A115361 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions