A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).
1, 466, 9681, 123106, 487345, 4511346, 6722401, 31576546, 63779361, 227102770, 235777201, 1191789186, 883674961, 3132638866, 4717986945, 8084578786, 7411648321, 29721182226, 17926949521, 59995093570, 65079564081, 109872175666, 81870270241, 305692541826
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[p_, e_] := (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)
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PARI
J(n, k) = sumdiv(n, d, d^k*moebius(n/d)); a(n, k=4, m=4) = sumdiv(n, d, J(d, k)*sigma(d^m));
Formula
a(n) = Sum_{d|n} J_4(d) * sigma(d^4), where the Jordan totient function J_4(n) = A059377(n).
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(5) * zeta(9) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 - 1/p^8 - 1/p^9 + 1/p^10) = 1.83382546873826519758... . (End)