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a(n) = A003415(sigma(n)) - n, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.
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%I #13 May 07 2021 01:12:42

%S -1,-1,1,-3,0,10,5,0,-8,11,5,20,-4,30,29,-15,4,-2,5,21,59,38,21,68,

%T -24,15,41,64,2,126,49,19,79,47,77,-16,-16,54,53,83,0,230,5,80,26,110,

%U 65,80,-27,-16,105,25,28,190,101,188,119,65,33,272,-28,210,101,-63,59,318,5,97,203,314,85,47,-34,27,53,112,195

%N a(n) = A003415(sigma(n)) - n, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

%H Antti Karttunen, <a href="/A342926/b342926.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A342925(n) - n = A003415(A000203(n)) - n.

%t Array[If[#2 < 2, 0, #2 Total[#2/#1 & @@@ FactorInteger[#2]]] - #1 & @@ {#, DivisorSigma[1, #]} &, 77] (* _Michael De Vlieger_, Apr 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 A342926(n) = (A003415(sigma(n))-n);

%Y Cf. A342925, A342924, A343223 [= gcd(A003415(n), a(n))].

%Y Cf. A342021 (positions of 0's), A343216 (of negative terms), A343217 (of nonnegative terms), A343218 (of positive terms).

%K sign

%O 1,4

%A _Antti Karttunen_, Apr 08 2021