Amiram Eldar, <a href="/A369716/b369716_1.txt">Table of n, a(n) for n = 1..10000</a>

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From Vaclav Kotesovec, Jan 30 2024: (Start)

Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).

Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).

Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where

f(1) = Product_{primes p} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,

f'(1) = f(1) * Sum_{primes p} (6*p^2 - 9*p + 4) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 2.35603132119230949914708478515883136510141335620960622673206366...,

f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-p*(12*p^5 - 27*p^4 + 16*p^3 + 9*p^2 - 12*p + 3) * log(p)^2 / (p^4 - 3*p^2 + 3*p - 1)^2) = f'(1)^2/f(1) + f(1) * (-7.3049026768735124341194605967271037971153161932236518820258070165876...),

gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

Vaclav Kotesovec, <a href="/A369716/a369716.jpg">Graph - the asymptotic ratio (100000 terms)</a>

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Amiram Eldar, <a href="/A369716/b369716_1.txt">Table of n, a(n) for n = 1..10000</a>

Cf. A000005, A001694, A197863, A369717, A369718, A369719.

allocated for Amiram EldarThe number of divisors of the smallest powerful number that is a multiple of n.

1, 3, 3, 3, 3, 9, 3, 4, 3, 9, 3, 9, 3, 9, 9, 5, 3, 9, 3, 9, 9, 9, 3, 12, 3, 9, 4, 9, 3, 27, 3, 6, 9, 9, 9, 9, 3, 9, 9, 12, 3, 27, 3, 9, 9, 9, 3, 15, 3, 9, 9, 9, 3, 12, 9, 12, 9, 9, 3, 27, 3, 9, 9, 7, 9, 27, 3, 9, 9, 27, 3, 12, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3

1,2

a(n) = A000005(A197863(n)).

Multiplicative with a(p) = 3 and a(p^e) = e+1 for e >= 2.

a(n) >= A000005(n), with equality if and only if n is powerful (A001694).

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 1/p^s - 2/p^(2*s) + 1/p^(3*s)).

f[p_, e_] := If[e == 1, 3, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

(PARI) a(n) = vecprod(apply(x -> if(x == 1, 3, x+1), factor(n)[, 2]));

Cf. A000005, A001694, A197863, A369717, A369718.

allocated

nonn,easy,mult

Amiram Eldar, Jan 30 2024

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