A392087 The unitary totient of the smallest square divisible by n.

1, 3, 8, 3, 24, 24, 48, 15, 8, 72, 120, 24, 168, 144, 192, 15, 288, 24, 360, 72, 384, 360, 528, 120, 24, 504, 80, 144, 840, 576, 960, 63, 960, 864, 1152, 24, 1368, 1080, 1344, 360, 1680, 1152, 1848, 360, 192, 1584, 2208, 120, 48, 72, 2304, 504, 2808, 240, 2880

Offset

1,2

Links

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

Formula

a(n) = A047994(A053143(n)).

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

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 1/p^4 + 1/p^5 + 1/p^7 - 1/p^8) = 0.57579534714985053947... .

Mathematica

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

Crossrefs

Cf. A002117, A013662, A047994, A053143, A183700, A327171.

Sequence in context: A046970 A322360 A392089 * A390831 A058936 A280369

Adjacent sequences: A392084 A392085 A392086 * A392088 A392089 A392090

Keyword

nonn,mult,easy

Author

Amiram Eldar, Dec 30 2025

Status

approved