A360165 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares minus the sum of the square roots of the unitary divisors of n that are even squares.

1, 1, 1, -1, 1, 1, 1, 1, 4, 1, 1, -1, 1, 1, 1, -3, 1, 4, 1, -1, 1, 1, 1, 1, 6, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -4, 1, 1, 1, 1, 1, 1, 1, -1, 4, 1, 1, -3, 8, 6, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 4, -7, 1, 1, 1, -1, 1, 1, 1, 4, 1, 1, 6, -1, 1, 1, 1, -3, 10, 1

Offset

1,9

Comments

The unitary analog of A347176.

Links

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

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} (-1)^(d+1)*sqrt(d).

a(n) = A360164(n) - 2 * A360162(n).

Multiplicative with a(2^e) = 1 - 2^(e/2) if e is even and 1 otherwise, and for p > 2, a(p^e) = p^(e/2) + 1 if e is even and 1 if e is odd.

Dirichlet g.f.: (zeta(s)*zeta(2*s-1)/zeta(3*s-1))*(2^(3*s)-2^(s+2)+2)/(2^(3*s)-2).

Sum_{k=1..n} a(k) ~ (n/Pi^2)*(log(n) + 3*gamma - 1 + 4*log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

Mathematica

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

Prog

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

Crossrefs

Cf. A001620, A306016, A347176, A360162, A360164.

Sequence in context: A368331 A366145 A204160 * A336870 A184101 A347176

Adjacent sequences: A360162 A360163 A360164 * A360166 A360167 A360168

Keyword

sign,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved