A351303 a(n) = n^8 * Product_{p|n, p prime} (1 + 1/p^8).

1, 257, 6562, 65792, 390626, 1686434, 5764802, 16842752, 43053282, 100390882, 214358882, 431727104, 815730722, 1481554114, 2563287812, 4311744512, 6975757442, 11064693474, 16983563042, 25700065792, 37828630724, 55090232674, 78310985282, 110522138624, 152588281250

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• mult

Comments

Sum of the 8th powers of the divisor complements of the squarefree divisors of n.

Links

• Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008683 (mu).

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), this sequence (k=8), A351304 (k=9), A351305 (k=10).

Programs

Mathematica

f[p_, e_] := p^(8*e) + p^(8*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Feb 08 2022 *)

PARI

a(n)=sumdiv(n, d, moebius(n/d)^2*d^8);

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^8*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

Formula

a(n) = Sum_{d|n} d^8 * mu(n/d)^2.

a(n) = n^8 * Sum_{d|n} mu(d)^2 / d^8.

Multiplicative with a(p^e) = p^(8*e) + p^(8*e-8). - Sebastian Karlsson, Feb 08 2022

From Vaclav Kotesovec, Feb 12 2022: (Start)

Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(2*s).

Sum_{k=1..n} a(k) ~ n^9 * zeta(9) / (9 * zeta(18)) = 4331032831125 * n^9 * zeta(9) / (43867 * Pi^18).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^8/(p^16-1)) = 1.004062071480173688638170669970682370243496458304179434830922739661777... (End)

a(n) = J_16(n)/J_8(n) = J_16(n)/A069093(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 14 2022