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A204455
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Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.
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21
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1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 1, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 55, 7, 57, 29, 59, 15, 61, 31, 21, 1, 65, 33, 67, 17, 69, 35, 71, 3
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OFFSET
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1,3
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COMMENTS
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There are no odd primes dividing n iff n is a power of 2.
This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.
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LINKS
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FORMULA
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Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - R. J. Mathar, Jul 02 2013
a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - Amiram Eldar, Nov 19 2022
a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - Peter Bala, Feb 01 2024
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EXAMPLE
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a(5)=5 because 5 is a single odd prime.
a(9)=3 because 9=3*3 has as squarefree part 3.
a(1)=1 because 1 is a power of 2, having no odd primes as a factor.
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MAPLE
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local p;
numtheory[factorset](n) minus {2} ;
mul(p, p=%) ;
end proc:
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MATHEMATICA
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f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* Michael De Vlieger, Apr 08 2015 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k, 1] % 2, f[k, 1], 1)); } \\ Michel Marcus, Apr 07 2015
(PARI) a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ Jianing Song, Aug 09 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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STATUS
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approved
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