A344178 - OEIS

%I #27 Dec 23 2024 09:51:55

%S 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 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 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 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

%C Question: Are all terms nonnegative? See also A211991 and A344584.

%C From _Bernard Schott_, May 25 2021: (Start)

%C 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 a(n) = 0 iff n is in A008578 (1 together with the primes).

%C a(n) = 1 iff n is in A006881 (squarefree semiprimes) (End).

%H Antti Karttunen, <a href="/A344178/b344178.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A344178/a344178.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = A003415(n) - A051953(n) = A168036(n) + A000010(n).

%t Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] - # + EulerPhi[#] &, 96] (* _Michael De Vlieger_, May 24 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 A344178(n) = A003415(n) - (n-eulerphi(n));

%Y Cf. A000010, A003415, A051953, A168036, A344584 (inverse Möbius transform).

%Y Cf. also A211991.

%Y Cf. A006881, A008578.

%K nonn

%O 1,4

%A _Antti Karttunen_, May 23 2021