A353961 - OEIS

%I #16 Jan 22 2025 17:32:23

%S 0,-1,-1,3,-1,-7,-1,15,5,-9,-1,13,-1,-11,6,47,-1,-22,-1,19,8,-15,-1,

%T 69,9,-17,-22,25,-1,-38,-1,127,12,-21,10,58,-1,-23,14,99,-1,-48,-1,37,

%U -27,-27,-1,213,13,-44,18,43,-1,-130,14,129,20,-33,-1,98,-1,-35,-37,319,16,-68

%N a(n) = Sum_{d|n} d' * (-1)^(d'), where d' is the arithmetic derivative of d (A003415).

%H Antti Karttunen, <a href="/A353961/b353961.txt">Table of n, a(n) for n = 1..20000</a>

%t d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#] * (-1)^d[#] &]; Array[a, 100] (* _Amiram Eldar_, May 12 2022 *)

%o (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415

%o a(n) = sumdiv(n, d, my(dd=ad(d)); dd*(-1)^dd); \\ _Michel Marcus_, May 13 2022

%Y Cf. A003415 (n').

%K sign

%O 1,4

%A _Wesley Ivan Hurt_, May 12 2022