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%I #10 Oct 16 2021 20:35:43
%S 0,5,34,1252,3711,270605,937924,225978828,2578767630,76913580247,
%T 310989720966,154447492062352,326011399456939,107109225555230817,
%U 3735502983248963648,627025327447691781152,875711370981239308953,872019874471537450417269,2082225625247428808306410
%N a(n) = Sum_{k=1..n} (n^k)' where ' is the arithmetic derivative.
%e a(4) = 1252; a(4) = Sum_{k=1..4} (4^k)' = (4^1)' + (4^2)' + (4^3)' + (4^4)' = 4' + 16' + 64' + 256' = 4 + 32 + 192 + 1024 = 1252.
%p a:= n-> add(n^k*add(i[2]/i[1], i=ifactors(n^k)[2]), k=1..n):
%p seq(a(n), n=1..19); # _Alois P. Heinz_, Oct 15 2021
%t d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := Sum[d[n^k], {k, 1, n}]; Array[a, 20] (* _Amiram Eldar_, Oct 16 2021 *)
%Y Cf. A003415.
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, Oct 15 2021