A390756 The Euler totient of the smallest cubefull number divisible by n.

1, 4, 18, 4, 100, 72, 294, 4, 18, 400, 1210, 72, 2028, 1176, 1800, 8, 4624, 72, 6498, 400, 5292, 4840, 11638, 72, 100, 8112, 18, 1176, 23548, 7200, 28830, 16, 21780, 18496, 29400, 72, 49284, 25992, 36504, 400, 67240, 21168, 77658, 4840, 1800, 46552, 101614, 144

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000010(A356193(n)).

Multiplicative with a(p^e) = (p-1) * p^2 for e < 3, and a(p^e) = (p-1) * p^(e-1) for e >= 3.

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

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * Product_{p prime} (1 - 2/p^2 + 3/p^5 - 3/p^6 + 1/p^7 - 1/p^8 + 3/p^9 - 3/p^10 + 1/p^11) = 0.43509138055168226516... .

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/ p^2 + 3/p^3 + 5/p^4 - 1/p^5 - 2/p^6) = 2.50811475105884725608... .

Mathematica

f[p_, e_] := (p-1) * p^If[e < 3, 2, e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1] - 1) * f[i, 1]^if(f[i, 2] < 3, 2, f[i, 2] - 1)); }

Crossrefs

Cf. A000010, A002117, A013661, A356193, A390755.

Sequence in context: A246133 A205014 A204936 * A390754 A158320 A342153

Adjacent sequences: A390753 A390754 A390755 * A390757 A390758 A390759

Keyword

nonn,mult,easy

Author

Amiram Eldar, Nov 17 2025

Status

approved