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

Author

Vladeta Jovovic, Feb 23 2004

Keywords

• mult
• easy
• nonn

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

Crossrefs

Cf. A056671, A092261.

Programs

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*)

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)