A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(u*v) = a(u)*v + u*a(v), where pi = A000720.

0, 0, 1, 2, 4, 3, 7, 4, 12, 12, 11, 5, 20, 6, 15, 19, 32, 7, 33, 8, 32, 26, 21, 9, 52, 30, 25, 54, 44, 10, 53, 11, 80, 37, 31, 41, 84, 12, 35, 44, 84, 13, 73, 14, 64, 87, 41, 15, 128, 56, 85, 55, 76, 16, 135, 58, 116, 62, 49, 17, 136, 18, 53, 120, 192, 69, 107

Offset: 0

Author

Alois P. Heinz, Jun 12 2015

Keywords

• nonn
• look

Comments

The pi-based variant of the arithmetic derivative of n (A003415).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Wikipedia, Arithmetic derivative
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Column k=1 of A258850, A258997.

First differences give A258863.

Partial sums give A258864.

Cf. A000720 (pi), A003415, A020639, A032742, A055396, A056239, A258861, A258862, A258995, A278510.

Programs

Maple

with(numtheory):
a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..100);

Mathematica

a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2016 *)

PARI

A258851(n)=n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]) \\ M. F. Hasler, Jul 13 2015

Scheme

(define (A258851 n) (if (<= n 1) 0 (+ (* (A055396 n) (A032742 n)) (* (A020639 n) (A258851 (A032742 n)))))) ;; Antti Karttunen, Mar 07 2017

Formula

a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.

a(A258861(n)) = n; A258861 = pi-based antiderivative of n.

a(a(A258862(n))) = n; A258862 = second pi-based antiderivative of n.

a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.

a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.

a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.

a(u^v) = v * u^(v-1) * a(u).