A351302 a(n) = n^7 * Product_{p|n, p prime} (1 + 1/p^7).

1, 129, 2188, 16512, 78126, 282252, 823544, 2113536, 4785156, 10078254, 19487172, 36128256, 62748518, 106237176, 170939688, 270532608, 410338674, 617285124, 893871740, 1290016512, 1801914272, 2513845188, 3404825448, 4624416768, 6103593750, 8094558822, 10465136172

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• mult

Comments

Sum of the 7th 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), A069092.

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), this sequence (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

Programs

Mathematica

f[p_, e_] := p^(7*e) + p^(7*(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^7);

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^7*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022

Formula

a(n) = Sum_{d|n} d^7 * mu(n/d)^2.

a(n) = n^7 * Sum_{d|n} mu(d)^2 / d^7.

Multiplicative with a(p^e) = p^(7*e) + p^(7*e-7). - Sebastian Karlsson, Feb 08 2022

From Vaclav Kotesovec, Feb 12 2022: (Start)

Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(2*s).

Sum_{k=1..n} a(k) ~ n^8 * zeta(8) / (8 * zeta(16)) = 34459425 * n^8 / (28936 * Pi^8).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^7/(p^14-1)) = 1.008287998838997802253937842472728682107868602338715231926150271159410... (End)

a(n) = J_14(n) / J_7(n) = J_14(n) / A069092(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022