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A074816
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a(n) = 3^A001221(n) = 3^omega(n).
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12
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1, 3, 3, 3, 3, 9, 3, 3, 3, 9, 3, 9, 3, 9, 9, 3, 3, 9, 3, 9, 9, 9, 3, 9, 3, 9, 3, 9, 3, 27, 3, 3, 9, 9, 9, 9, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 3, 9, 3, 9, 9, 9, 3, 9, 9, 9, 9, 9, 3, 27, 3, 9, 9, 3, 9, 27, 3, 9, 9, 27, 3, 9, 3, 9, 9, 9, 9, 27, 3, 9, 3, 9, 3, 27, 9, 9, 9, 9, 3, 27, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9
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OFFSET
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1,2
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COMMENTS
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Old name was: a(n) = sum(d|n, tau(d)*mu(d)^2 ).
Terms are powers of 3.
The inverse Mobius transform of A074823, as the Dirichlet g.f. is product_{primes p} (1+2*p^(-s)) and the Dirichlet g.f. of A074816 is product_{primes p} (1+2*p^(-s))/(1-p^(-s)). - R. J. Mathar, Feb 09 2011
If n is squarefree, then a(n) = #{(x, y) : x, y positive integers, lcm (x, y) = n}. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016
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REFERENCES
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R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.3 p. 108.
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LINKS
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FORMULA
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a(n) = 3^m if n is divisible by m distinct primes. i.e., a(n)=3 if n is in A000961; a(n)=9 if n is in A007774 ...
a(n) = Sum_{d|n, gcd(d, n/d) = 1} 2^omega(d). - Amiram Eldar, May 29 2020
Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Feb 16 2022
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MATHEMATICA
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PROG
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(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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