A347149 - OEIS

%I #24 Dec 25 2022 02:11:31

%S 1,3,3,0,3,9,3,0,0,9,3,0,3,9,9,0,3,0,3,0,9,9,3,0,0,9,0,0,3,27,3,0,9,9,

%T 9,0,3,9,9,0,3,27,3,0,0,9,3,0,0,0,9,0,3,0,9,0,9,9,3,0,3,9,0,0,9,27,3,

%U 0,9,27,3,0,3,9,0,0,9,27,3,0,0,9,3,0,9,9,9,0,3,0,9,0,9,9,9,0,3,0,0,0

%N Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

%H Vaclav Kotesovec, <a href="/A347149/a347149.jpg">Graph - the asymptotic ratio (1000000 terms)</a>.

%F Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)).

%F Let f(s) = Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)), then 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 - 6/p^2 + 8/p^3 - 3/p^4) = 0.1148840440802287887292512767015990978487135526872830176248484270625666728..., f'(1) = f(1) * Sum_{primes p} 12*log(p) / ((p-1)*(p+3)) = 0.5497153490016133577871571904347511299324572220423331992393596243955677299..., f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-24*p*(p-1) * log(p)^2 / ((p-1)^2 * (p+3)^2)) = 0.9028322988288094236586622799305270026576436536391185119652318723470259904... and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633).

%F a(n) = A008966(n) * A048691(n). - _Enrique Pérez Herrero_, Oct 27 2022

%F Multiplicative with a(p) = 3, and a(p^e) = 0 for e >= 2. - _Amiram Eldar_, Dec 25 2022

%t Table[MoebiusMu[n]^2 * 3^PrimeNu[n], {n, 1, 100}]

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X))[n], ", "))

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^3)[n], ", "))

%Y Cf. A008966, A074823, A165824.

%Y Cf. A013928, A069201, A048691.

%Y Cf. A001620, A082633.

%K nonn,mult

%O 1,2

%A _Vaclav Kotesovec_, Aug 20 2021