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A327928 Number of distinct primes p such that p^p divides the arithmetic derivative of n.

%I A327928 #10 Oct 01 2019 19:50:42
%S A327928 0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,
%T A327928 0,1,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0,0,0,
%U A327928 1,0,0,0,1,0,0,0,1,0,0,0,1,2,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0,0,1,0
%N A327928 Number of distinct primes p such that p^p divides the arithmetic derivative of n.
%H A327928 Antti Karttunen, <a href="/A327928/b327928.txt">Table of n, a(n) for n = 0..65537</a>
%F A327928 a(0) = a(1) = 0; for n > 1, a(n) = A129251(A003415(n)).
%e A327928 For n=20, A003415(20) = 24 = 2^3 * 3^1, thus only 2^2 divides 24, and a(24) = 1.
%e A327928 For n=81, A003415(81) = 108 = 2^2 * 3^3. Both 2^2 and 3^3 divide 108, thus a(81) = 2.
%o A327928 (PARI)
%o A327928 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A327928 A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
%o A327928 A327928(n) = if(n<=1,0,A129251(A003415(n)));
%Y A327928 Cf. A003415, A129251, A327929 (indices of nonzero terms), A327932.
%K A327928 nonn
%O A327928 0,82
%A A327928 _Antti Karttunen_, Oct 01 2019