A391897 Partial products of A063659.

1, 2, 6, 18, 90, 540, 3780, 22680, 181440, 1814400, 19958400, 179625600, 2335132800, 32691859200, 490377888000, 5884534656000, 100037089152000, 1600593426432000, 30411275102208000, 456169126533120000, 9579551657195520000, 210750136458301440000, 4847253138540933120000

Offset

1,2

Comments

a(n) is the determinant of the symmetric n X n matrix M defined by M(i,j) = psi(gcd(i,j)) for 1 <= i,j <= n, where psi(n) is the Dedekind psi function (A001615).

Links

Amiram Eldar, Table of n, a(n) for n = 1..456

Formula

a(n) = Product_{k=1..n} A063659(k).

Limit_{n->oo} a(n)^(1/n)/n = (1/e) * Product_{p prime} (1-1/p^2)^(1/p^2) = 0.33716171551022527677... . This constant equals exp(-c), where c = 1 - Sum_{p prime} log(1 - 1/p^2)/p^2 = 1 + Sum_{k>=2} P(2*k)/(k-1) = 1.087192595746731153648..., and P is the prime zeta function.

Mathematica

f[p_, e_] := If[e == 1, p, p^e - p^(e-2)]; A063659[n_] := Times @@ f @@@ FactorInteger[n]; FoldList[Times, Array[A063659, 25]]

Prog

(PARI) A063659(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], (f[i, 1]^2 - 1)*f[i, 1]^(f[i , 2] - 2))); }
list(nmax) = {my(p = 1); for(n = 1, nmax, p *= A063659(n); print1(p, ", ")); }

Crossrefs

Cf. A001615, A063659.

Similar sequences: A001088, A059381, A059382, A059383, A059384, A066780, A066843, A130779, A175596, A175836, A239672, A321613, A322175.

Sequence in context: A118455 A165774 A053505 * A000138 A028857 A052687

Adjacent sequences: A391894 A391895 A391896 * A391898 A391899 A391900

Keyword

nonn,easy

Author

Amiram Eldar, Dec 23 2025

Status

approved