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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.

%I A258851 #39 Mar 07 2017 13:16:30
%S A258851 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,
%T A258851 54,44,10,53,11,80,37,31,41,84,12,35,44,84,13,73,14,64,87,41,15,128,
%U A258851 56,85,55,76,16,135,58,116,62,49,17,136,18,53,120,192,69,107
%N 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.
%C A258851 The pi-based variant of the arithmetic derivative of n (A003415).
%H A258851 Alois P. Heinz, <a href="/A258851/b258851.txt">Table of n, a(n) for n = 0..10000</a>
%H A258851 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic derivative</a>
%H A258851 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A258851 a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.
%F A258851 a(A258861(n)) = n; A258861 = pi-based antiderivative of n.
%F A258851 a(a(A258862(n))) = n; A258862 = second pi-based antiderivative of n.
%F A258851 a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.
%F A258851 a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.
%F A258851 a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.
%F A258851 a(u^v) = v * u^(v-1) * a(u).
%p A258851 with(numtheory):
%p A258851 a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A258851 seq(a(n), n=0..100);
%t A258851 a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* _Jean-François Alcover_, Apr 24 2016 *)
%o A258851 (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
%o A258851 (Scheme) (define (A258851 n) (if (<= n 1) 0 (+ (* (A055396 n) (A032742 n)) (* (A020639 n) (A258851 (A032742 n)))))) ;; _Antti Karttunen_, Mar 07 2017
%Y A258851 Column k=1 of A258850, A258997.
%Y A258851 First differences give A258863.
%Y A258851 Partial sums give A258864.
%Y A258851 Cf. A000720 (pi), A003415, A020639, A032742, A055396, A056239, A258861, A258862, A258995, A278510.
%K A258851 nonn,look
%O A258851 0,4
%A A258851 _Alois P. Heinz_, Jun 12 2015