A360160 a(n) is the sum of unitary divisors of n that are odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 50, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 82, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Comments

The unitary analog of A360159.

Links

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

Crossrefs

Cf. A006519, A016754, A078434, A108269, A247041, A358347, A360159.

Programs

Mathematica

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

PARI

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

Formula

a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} d.

a(n) = A358347(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269), and otherwise it equals A358347(n)/(A006519(n)+1).

Multiplicative with a(2^e) = 1, 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+2))/(2^(3*s)-4).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (2*sqrt(2)/(4*sqrt(2)-1)) * zeta(3/2)/(3*zeta(5/2)) = 0.3942576405... .