This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386624 #17 Jul 28 2025 20:17:52
%S A386624 1,2,7,11,23,14,47,46,70,46,119,77,167,94,161,188,287,140,359,253,329,
%T A386624 238,527,322,596,334,642,517,839,322,959,760,833,574,1081,770,1367,
%U A386624 718,1169,1058,1679,658,1847,1309,1610,1054,2207,1316,2346,1192,2009,1837,2807,1284,2737,2162,2513,1678,3479,1771,3719,1918,3290,3056,3841,1666,4487,3157,3689
%N A386624 a(n) = Sum_{d|n} sigma(d) * phi(d) * mu(n/d).
%C A386624 Möbius transform of sigma(n) * phi(n) = A062354(n).
%H A386624 Amiram Eldar, <a href="/A386624/b386624.txt">Table of n, a(n) for n = 1..10000</a>
%F A386624 a(n) = Sum_{d|n} A062354(d) * mu(n/d).
%F A386624 From _Amiram Eldar_, Jul 27 2025: (Start)
%F A386624 Multiplicative with a(p) = p^2 - 2, and a(p^e) = p^(2*e) - p^(2*e-2) - p^(e-1) + p^(e-2) for e >= 2.
%F A386624 Dirichlet g.f.: (zeta(s-2) * zeta(s-1) / zeta(s)) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s + 1/p^(2*s-2)).
%F A386624 Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523*zeta(2)/(3*zeta(3)) = 0.24444595409976589792... . (End)
%t A386624 Table[Sum[EulerPhi[d] DivisorSigma[1, d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 100}]
%t A386624 f[p_, e_] := p^(2*e) - p^(e-1) - If[e > 1, p^(2*e-2) - p^(e-2), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 27 2025 *)
%o A386624 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i, 2]; p^(2*e) - p^(e - 1) - if(e > 1, p^(2*e - 2) - p^(e - 2), 1));} \\ _Amiram Eldar_, Jul 27 2025
%Y A386624 Cf. A000010 (phi), A000203 (sigma), A062354, A330523.
%K A386624 nonn,easy,mult
%O A386624 1,2
%A A386624 _Wesley Ivan Hurt_, Jul 27 2025