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A229347 a(1) = 1, for n > 1 a(n) = n*2^(omega(n)-1) where omega is A001221. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
If n, m > 1 and gcd(m,n) = 1 then a(n*m) = 2*a(n)*a(m).
LINKS
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
Sequence in context: A065636 A353694 A328260 * A367167 A135323 A052106
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 26 18:12 EDT 2024. Contains 372840 sequences. (Running on oeis4.)