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A055076
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Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.
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10
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1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 4, 8
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OFFSET
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1,2
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COMMENTS
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Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
The number of infinitary divisors of n that are squarefree (A005117). - Amiram Eldar, Jan 09 2024
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LINKS
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FORMULA
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a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - Antti Karttunen, Apr 05 2021
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
and gamma is the Euler-Mascheroni constant A001620. (End)
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EXAMPLE
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n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.
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MAPLE
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with(numtheory):
a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
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MATHEMATICA
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a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)
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PROG
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(PARI) A055076(n) = if(1==n, n, my(es=factor(n)[, 2]~); prod(i=1, #es, 2^(es[i]%2))); \\ Antti Karttunen, Apr 05 2021
(Scheme, with memoization-macro definec)
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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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