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A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.

%I A349172 #12 Nov 15 2021 12:03:55
%S A349172 1,6,8,24,12,48,16,84,44,72,24,192,28,96,96,276,36,264,40,288,128,144,
%T A349172 48,672,102,168,212,384,60,576,64,876,192,216,192,1056,76,240,224,
%U A349172 1008,84,768,88,576,528,288,96,2208,184,612,288,672,108,1272,288,1344,320,360,120,2304,124,384,704,2724,336,1152,136
%N A349172 a(n) = Sum_{d|n} psi(d) * A003959(n/d), where A003959 is fully multiplicative with a(p) = (p+1), and psi is Dedekind psi function, A001615.
%C A349172 Dirichlet convolution of A001615 with A003959.
%H A349172 Antti Karttunen, <a href="/A349172/b349172.txt">Table of n, a(n) for n = 1..20000</a>
%F A349172 a(n) = Sum_{d|n} A001615(d) * A003959(n/d).
%F A349172 a(n) = A327251(n) + A349142(n).
%F A349172 For all n >= 1, a(n) >= A349132(n).
%F A349172 Multiplicative with a(p^e) = (p+2)*(p+1)^e - (p+1)*p^e. - _Amiram Eldar_, Nov 09 2021
%t A349172 f[p_, e_] := (p + 2)*(p + 1)^e - (p + 1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *)
%o A349172 (PARI)
%o A349172 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
%o A349172 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349172 A349172(n) = sumdiv(n,d,A001615(d)*A003959(n/d));
%Y A349172 Cf. A001615, A003959, A327251, A349142, A349170, A349171, A349172, A349173.
%K A349172 nonn,mult
%O A349172 1,2
%A A349172 _Antti Karttunen_, Nov 09 2021