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A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).

%I A068719 #25 Apr 11 2025 03:32:11
%S A068719 1,4,5,12,7,16,9,32,21,24,13,44,15,32,31,80,19,60,21,68,41,48,25,112,
%T A068719 45,56,81,92,31,92,33,192,61,72,59,156,39,80,71,176,43,124,45,140,123,
%U A068719 96,49,272,77,140,91,164,55,216,87,240,101,120
%N A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).
%C A068719 Terms are either odd or multiples of 4. - _Antti Karttunen_, Jul 31 2022
%H A068719 Antti Karttunen, <a href="/A068719/b068719.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Bruno Berselli)
%F A068719 a(n) = A003415(A005843(n)).
%t A068719 ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[2*n]; Array[a, 100] (* _Amiram Eldar_, Apr 11 2025 *)
%o A068719 (Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [Ad(2*n): n in [1..60]]; // _Bruno Berselli_, Oct 22 2013
%o A068719 (PARI)
%o A068719 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A068719 A068719(n) = (n + 2*A003415(n)); \\ _Antti Karttunen_, Jul 31 2022
%Y A068719 Cf. A003415, A005843, A068720, A068721, A260624, A327862, A327864.
%Y A068719 Second diagonal (without the initial 1) in A084890.
%Y A068719 Row 1 of A344027.
%K A068719 nonn,easy
%O A068719 1,2
%A A068719 _Reinhard Zumkeller_, Feb 26 2002