A392088 The unitary totient of the smallest cube divisible by n.

1, 7, 26, 7, 124, 182, 342, 7, 26, 868, 1330, 182, 2196, 2394, 3224, 63, 4912, 182, 6858, 868, 8892, 9310, 12166, 182, 124, 15372, 26, 2394, 24388, 22568, 29790, 63, 34580, 34384, 42408, 182, 50652, 48006, 57096, 868, 68920, 62244, 79506, 9310, 3224, 85162, 103822

Offset

1,2

Links

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

Formula

a(n) = A047994(A053149(n)).

Multiplicative with a(p^e) = p^(e + ((3-e) mod 2)) - 1.

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

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(9) * Product_{p prime} (1 - 1/p^2 - 2/p^4 + 2/p^5 - 1/p^9 + 1/p^10 + 1/p^13 - 1/p^14) = 0.59019564513913623274... .

Mathematica

f[p_, e_] := p^(e + Mod[3-e, 3]) - 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]^(f[i, 2] + (3-f[i, 2])%3)-1); }

Crossrefs

Cf. A013662, A013667, A047994, A053149, A390754.

Sequence in context: A373162 A063453 A392090 * A284054 A284786 A262109

Adjacent sequences: A392085 A392086 A392087 * A392089 A392090 A392091

Keyword

nonn,mult,easy

Author

Amiram Eldar, Dec 30 2025

Status

approved