This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344178 #27 Dec 23 2024 09:51:55
%S A344178 0,0,0,2,0,1,0,8,3,1,0,8,0,1,1,24,0,9,0,12,1,1,0,28,5,1,18,16,0,9,0,
%T A344178 64,1,1,1,36,0,1,1,44,0,11,0,24,18,1,0,80,7,15,1,28,0,45,1,60,1,1,0,
%U A344178 48,0,1,24,160,1,15,0,36,1,13,0,108,0,1,20,40,1,17,0,128,81,1,0,64,1,1,1,92,0,57,1,48,1,1,1,208
%N A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).
%C A344178 Question: Are all terms nonnegative? See also A211991 and A344584.
%C A344178 From _Bernard Schott_, May 25 2021: (Start)
%C A344178 Answer: Yes, can be proved when n = Product_{i=1..k} p_i^e_i with n' = n * Sum_{i=1..k} (e_i/p_i) and cototient(n) = n * (1 - Product_{i=1..k} (1 - 1/p_i)).
%C A344178 a(n) = 0 iff n is in A008578 (1 together with the primes).
%C A344178 a(n) = 1 iff n is in A006881 (squarefree semiprimes) (End).
%H A344178 Antti Karttunen, <a href="/A344178/b344178.txt">Table of n, a(n) for n = 1..10000</a>
%H A344178 Antti Karttunen, <a href="/A344178/a344178.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A344178 a(n) = A003415(n) - A051953(n) = A168036(n) + A000010(n).
%t A344178 Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] - # + EulerPhi[#] &, 96] (* _Michael De Vlieger_, May 24 2021 *)
%o A344178 (PARI)
%o A344178 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A344178 A344178(n) = A003415(n) - (n-eulerphi(n));
%Y A344178 Cf. A000010, A003415, A051953, A168036, A344584 (inverse Möbius transform).
%Y A344178 Cf. also A211991.
%Y A344178 Cf. A006881, A008578.
%K A344178 nonn
%O A344178 1,4
%A A344178 _Antti Karttunen_, May 23 2021