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A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

%I A349143 #16 Nov 15 2021 01:32:15
%S A349143 0,1,1,9,1,16,1,51,13,22,1,114,1,28,25,233,1,145,1,168,31,40,1,590,21,
%T A349143 46,106,222,1,310,1,939,43,58,37,915,1,64,49,896,1,406,1,330,262,76,1,
%U A349143 2570,29,297,61,384,1,1012,49,1202,67,94,1,2040,1,100,340,3489,55,598,1,492,79,574,1,4457,1,118,360,546,55
%N A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).
%C A349143 Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
%C A349143 Dirichlet convolution of the identity function (A000027) with A349140.
%C A349143 Dirichlet convolution of sigma (A000203) with A349141.
%C A349143 Dirichlet convolution of A060640 with A348971.
%H A349143 Antti Karttunen, <a href="/A349143/b349143.txt">Table of n, a(n) for n = 1..16384</a>
%F A349143 a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
%F A349143 a(n) = Sum_{d|n} d * A349140(n/d).
%F A349143 a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
%F A349143 a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
%F A349143 For all n >= 1, a(n) >= A349123(n) >= A348983(n).
%t A349143 f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)
%o A349143 (PARI)
%o A349143 A038040(n) = (n*numdiv(n));
%o A349143 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349143 A348507(n) = (A003959(n) - n);
%o A349143 A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));
%Y A349143 Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.
%Y A349143 Cf. also A349123, A348983.
%K A349143 nonn
%O A349143 1,4
%A A349143 _Antti Karttunen_, Nov 08 2021