A335115 a(2*n) = 2*n - a(n), a(2*n+1) = 2*n + 1.

1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 9, 13, 7, 15, 11, 17, 9, 19, 15, 21, 11, 23, 15, 25, 13, 27, 21, 29, 15, 31, 21, 33, 17, 35, 27, 37, 19, 39, 25, 41, 21, 43, 33, 45, 23, 47, 33, 49, 25, 51, 39, 53, 27, 55, 35, 57, 29, 59, 45, 61, 31, 63, 43, 65, 33, 67, 51, 69, 35, 71, 45, 73, 37, 75

Offset: 1

Author

Ilya Gutkovskiy, May 23 2020

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A035263, A050292, A129527, A154269, A001045.

Programs

Mathematica

a[n_] := a[n] = If[EvenQ[n], n - a[n/2], n]; Table[a[n], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(-1)^k x^(2^k)/(1 - x^(2^k))^2, {k, 0, Floor[Log[2, nmax]]}], {x, 0, nmax}], x] // Rest
f[p_, e_] := If[p == 2, (2^(e + 1) + (-1)^e)/3, p^e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

PARI

a(n) = my(k=valuation(n,2)); (n<<1 + (n>>k)*(-1)^k)/3; \\ Kevin Ryde, Oct 06 2020

Formula

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

G.f. A(x) satisfies: A(x) = x / (1 - x)^2 - A(x^2).

Dirichlet g.f.: zeta(s-1) / (1 + 2^(-s)).

a(n) = Sum_{d|n} A154269(n/d) * d.

Sum_{k=1..n} a(k) ~ 2*n^2/5. - Vaclav Kotesovec, Jun 11 2020

Multiplicative with a(2^e) = A001045(e+1) and a(p^e) = p^e for e >= 0 and prime p > 2. - Werner Schulte, Oct 05 2020