A360161 a(n) is the sum of unitary divisors of n that are odd squares minus the sum of unitary divisors of n that are even squares.

1, 1, 1, -3, 1, 1, 1, 1, 10, 1, 1, -3, 1, 1, 1, -15, 1, 10, 1, -3, 1, 1, 1, 1, 26, 1, 1, -3, 1, 1, 1, 1, 1, 1, 1, -30, 1, 1, 1, 1, 1, 1, 1, -3, 10, 1, 1, -15, 50, 26, 1, -3, 1, 1, 1, 1, 1, 1, 1, -3, 1, 1, 10, -63, 1, 1, 1, -3, 1, 1, 1, 10, 1, 1, 26, -3, 1, 1, 1

Offset

1,4

Comments

The unitary analog of A344300.

Links

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

Formula

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

a(n) = A360160(n) - 2 * A358347(n).

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

Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+3)+4)/(2^(3*s)-4).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)*(4*sqrt(2)-1)) = 0.1393911255... .

Mathematica

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

Crossrefs

Cf. A016742, A016754, A078434, A247041, A344300, A358347, A360160.

Sequence in context: A360440 A349350 A061494 * A141901 A200473 A361948

Adjacent sequences: A360158 A360159 A360160 * A360162 A360163 A360164

Keyword

sign,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved