A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

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, 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, 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

Offset

1,4

Comments

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

From Bernard Schott, May 25 2021: (Start)

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)).

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

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

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

Mathematica

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] - # + EulerPhi[#] &, 96] (* Michael De Vlieger, May 24 2021 *)

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A344178(n) = A003415(n) - (n-eulerphi(n));

Crossrefs

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

Cf. also A211991.

Cf. A006881, A008578.

Sequence in context: A185415 A049218 A212358 * A357078 A154469 A344584

Adjacent sequences: A344175 A344176 A344177 * A344179 A344180 A344181

Keyword

nonn,changed

Author

Antti Karttunen, May 23 2021

Status

approved