A091306 Sum of squares of unitary, squarefree divisors of n, including 1.

1, 5, 10, 1, 26, 50, 50, 1, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 10, 1, 850, 1, 50, 842, 1300, 962, 1, 1220, 1450, 1300, 1, 1370, 1810, 1700, 26, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 5, 3172, 50, 3620

Offset

1,2

Comments

If b(n,k) = sum of k-th powers of unitary, squarefree divisors of n, including 1, then b(n,k) is multiplicative with b(p,k)=p^k+1 and b(p^e,k)=1 for e>1.

Dirichlet g.f.: zeta(s)*product_{primes p} (1+p^(2-s)-p^(2-2s)). Dirichlet convolution of A000012 with the multiplicative sequence 1, 4, 9, -4, 25, 36, 49, 0, -9, 100, 121, -36, 169, 196,... - R. J. Mathar, Aug 28 2011

Links

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

Formula

Multiplicative with a(p)=p^2+1 and a(p^e)=1 for e>1.

From Vaclav Kotesovec, Nov 20 2021: (Start)

Dirichlet g.f.: zeta(s) * zeta(s-2) * Product_{primes p} (1 + p^(4 - 3*s) - p^(2 - 2*s) - p^(4 - 2*s)).

Sum_{k=1..n} a(k) ~ c * zeta(3) * n^3 / 3, where c = Product_{primes p} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.576152735385667059520611078264117275406247116802896188...

(End)

Mathematica

f[p_, e_] := If[e == 1, p^2 + 1, 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 30 2019*)

Crossrefs

Cf. A056671, A092261.

Sequence in context: A330599 A099731 A307716 * A073048 A102258 A361672

Adjacent sequences: A091303 A091304 A091305 * A091307 A091308 A091309

Keyword

mult,easy,nonn

Author

Vladeta Jovovic, Feb 23 2004

Status

approved