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A362632 a(n) = Sum_{d|n, gcd(d,n/d)=1} d * psi(d), where psi is the Dedekind psi function (A001615).

%I A362632 #15 Feb 16 2025 08:34:05
%S A362632 1,7,13,25,31,91,57,97,109,217,133,325,183,399,403,385,307,763,381,
%T A362632 775,741,931,553,1261,751,1281,973,1425,871,2821,993,1537,1729,2149,
%U A362632 1767,2725,1407,2667,2379,3007,1723,5187,1893,3325,3379,3871,2257,5005,2745,5257,3991,4575,2863
%N A362632 a(n) = Sum_{d|n, gcd(d,n/d)=1} d * psi(d), where psi is the Dedekind psi function (A001615).
%H A362632 Amiram Eldar, <a href="/A362632/b362632.txt">Table of n, a(n) for n = 1..10000</a>
%H A362632 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindFunction.html">Dedekind Function</a>.
%H A362632 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.
%F A362632 a(p) = p^2 + p + 1, p prime.
%F A362632 From _Amiram Eldar_, May 03 2023: (Start)
%F A362632 Multiplicative with a(p^e) = 1 + (p+1)*p^(2*e-1).
%F A362632 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p^4 + p^3 + 2*p^2 + 2*p + 1)/(p^2*(p^2 + p + 1)) = 0.55359070186594463118... . (End)
%t A362632 f[p_, e_] := 1 + (p + 1)*p^(2*e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 03 2023 *)
%Y A362632 Cf. A001615 (psi), A034444, A060648.
%K A362632 nonn,easy,mult
%O A362632 1,2
%A A362632 _Wesley Ivan Hurt_, Apr 28 2023