A097378 a(n) = SquareFreeKernel(n)*CubeFreeKernel(n) + 1.

2, 5, 10, 9, 26, 37, 50, 9, 28, 101, 122, 73, 170, 197, 226, 9, 290, 109, 362, 201, 442, 485, 530, 73, 126, 677, 28, 393, 842, 901, 962, 9, 1090, 1157, 1226, 217, 1370, 1445, 1522, 201, 1682, 1765, 1850, 969, 676, 2117, 2210, 73, 344, 501, 2602, 1353, 2810

Offset: 1

Author

Reinhard Zumkeller, Aug 11 2004

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Cubefree.
• Eric Weisstein's World of Mathematics, Squarefree.

Crossrefs

Cf. A002117, A004709, A005117, A007947, A007948, A097381.

Programs

Mathematica

f[p_, e_] := p^(1 + Min[e, 2]); a[1] = 2; a[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 01 2024 *)

PARI

a(n) = {my(f = factor(n)); 1 + prod(i = 1, #f~, f[i, 1]^(1 + min(f[i, 2], 2)));} \\ Amiram Eldar, Feb 01 2024

Formula

a(n) = A007947(n)*A007948(n) + 1.

From Amiram Eldar, Feb 01 2024: (Start)

b(n) = a(n) - 1 is multiplicative with b(p^e) = p^(1 + min(e, 2)).

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

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.69256837284462414024... . (End)