A229347 a(1) = 1, for n > 1 a(n) = n*2^(omega(n)-1) where omega is A001221.

1, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 120, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 168, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 240, 61, 124, 126

Offset

1,2

Comments

If n, m > 1 and gcd(m,n) = 1 then a(n*m) = 2*a(n)*a(m).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

For this sequence but with a(1)=0, Dirichlet g.f.: zeta(s-1)*primezeta(s-1) where primezeta(s) is Sum_{prime p} p^(-s). - Benedict W. J. Irwin, Jul 14 2018 [This g.f. is incorrect. With a(1)=0, Sum_{k>=1} a(k)/k^3 = 0.75, but zeta(2)*primezeta(2) = 0.74391718786976797493... - Michael Shamos, Mar 22 2023]

For n > 1, Dirichlet g.f. for 2*a(n) is zeta(s-1)^2/zeta(2*s-2). - Vaclav Kotesovec, Jan 10 2024

Sum_{k=1..n} a(k) ~ 3*n^2/Pi^2 * (log(n)/2 + gamma - 1/4 - 6*zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 23 2023

For n>1, a(n) = abs(A298473)/2. - Vaclav Kotesovec, Jan 10 2024

Mathematica

h[n_] := 2^(Length[FactorInteger[n]] - 1)*n; Array[h, 100]
Join[{1}, Table[n*2^(PrimeNu[n]-1), {n, 2, 70}]] (* Harvey P. Dale, Mar 13 2015 *)

Prog

(PARI) a(n)=n<<max(omega(n)-1, 0) \\ Charles R Greathouse IV, Sep 24 2013
(PARI) print1(1, ", "); for(n=2, 100, print1(1/2 * direuler(p=2, n, (1 - p^2*X^2) / (1-p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jan 10 2024

Crossrefs

Cf. A001221, A034444, A298473.

Sequence in context: A065636 A353694 A328260 * A367167 A135323 A052106

Adjacent sequences: A229344 A229345 A229346 * A229348 A229349 A229350

Keyword

nonn

Author

José María Grau Ribas, Sep 24 2013

Status

approved