A351301 a(n) = n^6 * Product_{p|n, p prime} (1 + 1/p^6).

1, 65, 730, 4160, 15626, 47450, 117650, 266240, 532170, 1015690, 1771562, 3036800, 4826810, 7647250, 11406980, 17039360, 24137570, 34591050, 47045882, 65004160, 85884500, 115151530, 148035890, 194355200, 244156250, 313742650, 387951930, 489424000, 594823322, 741453700

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• mult

Comments

Sum of the 6th 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), this sequence (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

Programs

Mathematica

f[p_, e_] := p^(6*e) + p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Feb 08 2022 *)

PARI

a(n)=sumdiv(n, d, moebius(n/d)^2*d^6);

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^6*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

Formula

a(n) = Sum_{d|n} d^6 * mu(n/d)^2.

a(n) = n^6 * Sum_{d|n} mu(d)^2 / d^6.

Multiplicative with a(p^e) = p^(6*e) + p^(6*e-6). - Sebastian Karlsson, Feb 08 2022

From Vaclav Kotesovec, Feb 12 2022: (Start)

Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(2*s).

Sum_{k=1..n} a(k) ~ n^7 * zeta(7) / (7 * zeta(14)) = 2606175 * n^7 * zeta(7) / (2 * Pi^14).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^6/(p^12-1)) = 1.01709659289559607702424749979498914920118274875188346777424441790304... (End)