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A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

%I A328768 #20 Jul 08 2024 13:59:28
%S A328768 0,0,1,2,4,6,7,30,12,12,17,210,20,2310,67,28,32,30030,33,510510,44,
%T A328768 104,431,9699690,52,60,4633,54,148,223092870,71,6469693230,80,652,
%U A328768 60077,192,84,200560490130,1021039,6956,108,7420738134810,229,304250263527210,884,114,19399403,13082761331670030,128,420,145,90124,9292,614889782588491410,135,1116,324
%N A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.
%H A328768 Antti Karttunen, <a href="/A328768/b328768.txt">Table of n, a(n) for n = 0..1001</a>
%H A328768 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A328768 a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).
%F A328768 a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).
%F A328768 For all n >= 0, A276150(a(n)) = A328771(n).
%o A328768 (PARI)
%o A328768 A002110(n) = prod(i=1,n,prime(i));
%o A328768 A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
%o A328768 (PARI) A328768(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]^2)));
%Y A328768 Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.
%Y A328768 Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].
%Y A328768 Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].
%Y A328768 For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.
%K A328768 nonn
%O A328768 0,4
%A A328768 _Antti Karttunen_, Oct 28 2019