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

1,9

Comments

The unitary analog of A360159.

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} 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... .

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]

Prog

(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))); }

Crossrefs

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

Sequence in context: A367614 A367616 A118762 * A360159 A284100 A279051

Adjacent sequences: A360157 A360158 A360159 * A360161 A360162 A360163

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 29 2023

Status

approved