A381713 a(n) = J_9(n)/J_3(n), where J_k is the k-th Jordan totient function.
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
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
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 *)
-
PARI
J(n, k) = sumdiv(n, d, d^k*moebius(n/d)); a(n) = J(n, 9)/J(n, 3);
-
PARI
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
Formula
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)