

A129527


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


18



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
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OFFSET

0,3


COMMENTS

Sum of odd part of n and its double, fourfold, eightfold etc. <= n.
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(n1) not including 2(n1) that are obtained dividing repeatedly by 2) + (greatest odd divisor of 2(n1), including 1), for the initial case 2(11)=0 will be set to 0. E.g., (offset is starting with n=1) a(3) = Sum(even divisors of 2*(31)=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*(41)=6 not including 6 obtained dividing repeatedly by 2) + greatest odd divisor of 6 = (0) + (3) = 3; a(5) = Sum(even divisors of 2*(51)=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



FORMULA

G.f.: Sum_{k>=0} x^(2^k)/(1x^(2^k))^2.
Dirichlet g.f.: zeta(s1)*2^s/(2^s1).  Ralf Stephan, Jun 17 2007
G.f. satisfies g(x) = g(x^2) + x/(1x)^2.  Robert Israel, Dec 20 2015
Conjecture: a(n) = 2*nA000265(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


MAPLE

f:= proc(n) option remember;
if n::odd then n else n + procname(n/2) fi
end proc:
f(0):= 0:


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)=if (n==0, 0, sum(k=0, valuation(n, 2), n/2^k)); \\ corrected by Michel Marcus, Dec 22 2021
(PARI) a(n)=sumdiv(n, d, eulerphi(2*d)) \\ Andrew Howroyd, Aug 07 2018


CROSSREFS



KEYWORD

nonn,easy,mult


AUTHOR



STATUS

approved



