A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 2, 4, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[p_, e_] := If[p <= 3, e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, f[i,2]+1, 2));}
Formula
Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.
Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).
Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).
Comments