A194532 -id:A194532 - cp's OEIS frontend

A381712 Euler transform of n * A194532(n).

1, 1, 43, 316, 2563, 17284, 135843, 903141, 6153645, 39839122, 256023118, 1589382754, 9751548710, 58451287319, 345478493273, 2006641555356, 11498560570683, 64940715401160, 362249937059777, 1995639600211016, 10870475203155005, 58563229198239242, 312277069694594537

Offset: 0

Seiichi Manyama, Mar 05 2025

nonn

Cf. A156304, A381714.

Cf. A194532.

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^3, 2)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^(k * A194532(k)).
G.f.: exp( Sum_{k>=1} sigma_2(k^3) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_2(k^3) * a(n-k).

A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

Original entry on oeis.org

1, 73, 757, 4672, 15751, 55261, 117993, 299008, 551853, 1149823, 1772893, 3536704, 4829007, 8613489, 11923507, 19136512, 24142483, 40285269, 47052741, 73588672, 89320701, 129421189, 148048057, 226349056, 246109375, 352517511, 402300837, 551263296

Offset: 1

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A065958, A160889, A194532.
Cf. A059376, A069094.
Cf. A013664, A013667.

f[p_, e_] := p^(6*e) * (1 + 1/p^3 + 1/p^6); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 28] (* Amiram Eldar, Mar 05 2025 *)

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n) = J(n, 9)/J(n, 3);

a(n) = {my(p = factor(n)[, 1]); n^6 * prod(i = 1, #p, 1 + 1/p[i]^3 + 1/p[i]^6);} \\ Amiram Eldar, Mar 05 2025

a(n) = A069094(n)/A059376(n).
a(n) = n^6 * Product_{distinct primes p dividing n} (1 + 1/p^3 + 1/p^6).
From Amiram Eldar, Mar 05 2025: (Start)
Dirichlet g.f.: zeta(s-6) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = Product_{p prime} (1 + 1/p^4 + 1/p^7) = 1.08635980686198102055... .
Sum_{n>=1} 1/a(n) = zeta(6)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^15) = 1.01533121878447451064... . (End)

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A381712 Euler transform of n * A194532(n).

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A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.

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