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A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

%I A353379 #12 Jan 29 2025 11:15:55
%S A353379 0,1,1,2,1,4,1,3,2,4,1,7,1,4,4,4,1,7,1,7,4,4,1,11,2,4,3,7,1,12,1,5,4,
%T A353379 4,4,12,1,4,4,11,1,12,1,7,7,4,1,15,2,7,4,7,1,11,4,11,4,4,1,20,1,4,7,6,
%U A353379 4,12,1,7,4,12,1,19,1,4,7,7,4,12,1,15,4,4,1,20,4,4,4,11,1,20,4,7,4,4,4,21,1,7,7
%N A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.
%H A353379 Antti Karttunen, <a href="/A353379/b353379.txt">Table of n, a(n) for n = 1..16384</a>
%H A353379 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A353379 a(n) = A258851(A181819(n)).
%p A353379 a:= n-> (m-> m*add(i[2]*numtheory[pi](i[1])/i[1], i=ifactors(m)[2]))
%p A353379         (mul(ithprime(i[2]), i=ifactors(n)[2])):
%p A353379 seq(a(n), n=1..120);  # _Alois P. Heinz_, Apr 28 2022
%t A353379 a[n_] := If[n == 1, 0, #*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[#]}]]&[Product[Prime[i[[2]]], {i, FactorInteger[n]}]];
%t A353379 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 29 2025, after _Alois P. Heinz_ *)
%o A353379 (PARI)
%o A353379 A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
%o A353379 A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
%o A353379 A353379(n) = A258851(A181819(n));
%Y A353379 Cf. A181819, A258851.
%Y A353379 Cf. also A351942.
%K A353379 nonn
%O A353379 1,4
%A A353379 _Antti Karttunen_, Apr 28 2022