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A293303
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Exponential convolution of the exponential Mobius function and the natural numbers.
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1
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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
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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s[p_, e_] := DivisorSum[e, MoebiusMu[e/#]*p^#&];
a[n_] := a[n] = Times @@ s @@@ FactorInteger[n];
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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