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%I #19 Nov 15 2021 03:40:27
%S 0,1,1,8,1,15,1,40,12,21,1,96,1,27,24,160,1,126,1,144,30,39,1,440,20,
%T 45,90,192,1,279,1,560,42,57,36,720,1,63,48,680,1,369,1,288,234,75,1,
%U 1680,28,270,60,336,1,810,48,920,66,93,1,1656,1,99,306,1792,54,549,1,432,78,531,1,3120,1,117,330,480,54,639
%N a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.
%C This sequence is the Dirichlet convolution of at least the following pairs of sequences:
%C - A003415 (the arithmetic derivative) with A038040,
%C - A000027 (the identity function) with A347130,
%C - A000203 (sigma) with A347131,
%C - A018804 with A319684,
%C - A060640 with A300251.
%H Antti Karttunen, <a href="/A349123/b349123.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = Sum_{d|n} A038040(d) * A003415(n/d).
%F a(n) = Sum_{d|n} d * A347130(n/d).
%F a(n) = Sum_{d|n} A000203(d) * A347131(n/d).
%F a(n) = Sum_{d|n} A018804(d) * A319684(n/d).
%F a(n) = Sum_{d|n} A060640(d) * A300251(n/d).
%F For all n >= 1, A348983(n) <= a(n) <= A349143(n).
%F a(n) = A003557(n) * A349124(n).
%t d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A038040(n) = (n*numdiv(n));
%o A349123(n) = sumdiv(n,d,A038040(d)*A003415(n/d));
%Y Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124.
%Y Cf. also A348983, A349143.
%K nonn
%O 1,4
%A _Antti Karttunen_, Nov 08 2021
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