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A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

%I A328769 #10 Oct 28 2019 20:01:09
%S A328769 0,0,2,6,8,30,18,210,24,36,70,2310,48,30030,434,120,64,510510,90,
%T A328769 9699690,160,672,4642,223092870,120,300,60086,162,896,6469693230,270,
%U A328769 200560490130,160,6996,1021054,1260,216,7420738134810,19399418,90168,360,304250263527210,1386,13082761331670030,9328,450,446185786,614889782588491410,288,2940,650,1531632
%N A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.
%H A328769 Antti Karttunen, <a href="/A328769/b328769.txt">Table of n, a(n) for n = 0..1001</a>
%H A328769 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A328769 a(n) = n * Sum e_j * (p_j)#/p_j for n = Product p_j^e_j with (p_j)# = A034386(p_j).
%F A328769 A276150(a(n)) = A328772(n).
%o A328769 (PARI)
%o A328769 A034386(n) = factorback(primes(primepi(n)));
%o A328769 A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A034386(f[i,1])/f[i, 1]));
%o A328769 (PARI)
%o A328769 A002110(n) = prod(i=1,n,prime(i));
%o A328769 A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1]))/f[i, 1]));
%Y A328769 Cf. A002110, A034386, A108951, A328772.
%Y A328769 Cf. also A003415, A258851, A328768, A328845, A328846.
%K A328769 nonn
%O A328769 0,3
%A A328769 _Antti Karttunen_, Oct 28 2019