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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
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
KEYWORD
nonn,easy,mult
AUTHOR
R. J. Mathar, Oct 05 2017
STATUS
approved