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A323308
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The number of exponential semiproper divisors of n.
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21
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1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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An exponential semiproper divisor of n is a divisor d such that rad(d) = rad(n) and gcd(d/rad(n), n/d) = 1, where rad(n) is the largest squarefree divisor of n (A007947).
a(n) is also the number of divisors of n that are squares of squarefree numbers (A062503). - Amiram Eldar, Oct 08 2022
a(n) is also the number of unitary divisors of n that are powerful (A001694). - Amiram Eldar, Feb 18 2023
The smallest integer that has exactly 2^n exponential semiproper divisors is A061742(n). - Bernard Schott, Feb 20 2023
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LINKS
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FORMULA
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Multiplicative with a(p^e) = 1 for e = 1 and 2 otherwise.
Asymptotic mean: Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 15/Pi^2 = 1.5198177546... (A082020). - Amiram Eldar, Nov 08 2020
Dirichlet g.f.: zeta(s) * zeta(2*s) / zeta(4*s). - Werner Schulte, Dec 29 2022
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MATHEMATICA
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f[p_, e_] := If[e==1, 1, 2]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1] = min(f[k, 2], 2); f[k, 2] = 1); factorback(f); \\ Michel Marcus, Jan 11 2019
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CROSSREFS
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Cf. A000188, A001694, A007947, A008833, A034444, A057521, A061742, A062503, A082020, A323309, A323310.
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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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