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

%I A129527 #80 Feb 24 2025 22:43:24
%S A129527 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,
%T A129527 39,27,49,29,45,31,63,33,51,35,63,37,57,39,75,41,63,43,77,45,69,47,93,
%U A129527 49,75,51,91,53,81,55,105,57,87,59,105,61,93,63,127,65,99,67
%N A129527 a(2n) = a(n) + 2n, a(2n+1) = 2n + 1.
%C A129527 Sum of odd part of n and its double, fourfold, eightfold etc. <= n.
%C A129527 Starting with 1 = the ruler function triangle A115361 * [1, 2, 3, ...]. - _Gary W. Adamson_, Nov 27 2009
%C A129527 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
%C A129527 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
%C A129527 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
%H A129527 Robert Israel, <a href="/A129527/b129527.txt">Table of n, a(n) for n = 0..10000</a>
%H A129527 Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.3390/sym16010005">Plane Partitions and Divisors</a>, Symmetry (2024), Vol. 16, Iss. 5. See page 3.
%H A129527 Mircea Merca and Emil Simion, <a href="https://doi.org/10.3390/sym15112067">n-Color Partitions into Distinct Parts as Sums over Partitions</a>, Symmetry (2023) Vol. 15, Iss. 11.
%F A129527 G.f.: Sum_{k>=0} x^(2^k)/(1-x^(2^k))^2.
%F A129527 Dirichlet g.f.: zeta(s-1)*2^s/(2^s-1). - _Ralf Stephan_, Jun 17 2007
%F A129527 G.f. satisfies g(x) = g(x^2) + x/(1-x)^2. - _Robert Israel_, Dec 20 2015
%F A129527 n <= a(n) <= 2n - 1 for n > 0. - _Charles R Greathouse IV_, Feb 09 2016
%F A129527 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.
%F A129527 _Andrey Zabolotskiy_, Aug 03 2017: Confirm: induction works, the conjecture holds for all n.]
%F A129527 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
%F A129527 Inverse Moebius transform of A062570. - _Andrew Howroyd_, Aug 07 2018
%F A129527 Sum_{k=1..n} a(k) ~ 2*n^2/3. - _Vaclav Kotesovec_, Jun 11 2020
%F A129527 a(n) = A038712(n)*A000265(n), for n > 0. - _Ivan N. Ianakiev_, Feb 24 2025
%p A129527 f:= proc(n) option remember;
%p A129527   if n::odd then n else n + procname(n/2) fi
%p A129527 end proc:
%p A129527 f(0):= 0:
%p A129527 seq(f(n),n=0..100); # _Robert Israel_, Dec 20 2015
%t A129527 a[n_] := a[n] = If[EvenQ@ n, a[n/2] + n, n]; {0}~Join~Array[a, 67] (* _Michael De Vlieger_, Jun 26 2017 *)
%o A129527 (PARI) a(n)=if (n==0, 0, sum(k=0,valuation(n,2),n/2^k)); \\ corrected by _Michel Marcus_, Dec 22 2021
%o A129527 (PARI) a(n)=if(n<2,return(n)); my(k=valuation(n,2)); 2*n-n>>k \\ _Charles R Greathouse IV_, Feb 09 2016
%o A129527 (PARI) a(n)=sumdiv(n, d, eulerphi(2*d)) \\ _Andrew Howroyd_, Aug 07 2018
%Y A129527 Row sums of A129265 and A129559.
%Y A129527 Cf. A000265, A006516, A062570, A115361.
%K A129527 nonn,easy,mult
%O A129527 0,3
%A A129527 _Ralf Stephan_, May 29 2007