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A293303 Exponential convolution of the exponential Mobius function and the natural numbers. 1
1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 12, 17, 12, 19, 10, 21, 22, 23, 18, 20, 26, 24, 14, 29, 30, 31, 30, 33, 34, 35, 12, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 36, 42, 40, 51, 26, 53, 48, 55, 42, 57, 58, 59, 30, 61, 62, 42, 54, 65, 66, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Exponential convolution of A166234 and A000027.
Similar to the definition of A000010 as the Dirichlet convolution of A008683 and A000027.
LINKS
Andrew V. Lelechnko, Exponenital and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence f^E(n).
FORMULA
Multiplicative with a(p^e) = Sum_{d|e} A008683(e/d)*p^d.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.43802998037163511363... = (1/2) * Product_{p prime} (1-1/p)*Sum_{k>=1} (Sum_{d|e} mu(k/d)*p^k/p^(2*k)). - Amiram Eldar, Oct 03 2023
MAPLE
A293303 := proc(n)
local a, pe, i, p, e, f, d ;
a := 1 ;
for pe in ifactors(n)[2] do
p := pe[1] ;
e := pe[2] ;
f := 0 ;
for d in numtheory[divisors](e) do
f := f+numtheory[mobius](e/d)*p^d ;
end do:
a := a*f ;
end do:
a ;
end proc:
seq(A293303(n), n=1..100) ;
MATHEMATICA
s[p_, e_] := DivisorSum[e, MoebiusMu[e/#]*p^#&];
a[n_] := a[n] = Times @@ s @@@ FactorInteger[n];
Array[a, 100] (* Jean-François Alcover, Dec 09 2017 *)
CROSSREFS
Sequence in context: A309108 A308056 A336965 * A333569 A110500 A161871
KEYWORD
nonn,easy,mult
AUTHOR
R. J. Mathar, Oct 05 2017
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)