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A327932 a(n) = A327928(n) - A129251(n), where A327928(n) gives the number of distinct primes p such that p^p divides the arithmetic derivative of n, and A129251(n) gives the number of such primes for n.

%I A327932 #9 Jul 08 2020 21:52:24
%S A327932 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A327932 1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A327932 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0
%N A327932 a(n) = A327928(n) - A129251(n), where A327928(n) gives the number of distinct primes p such that p^p divides the arithmetic derivative of n, and A129251(n) gives the number of such primes for n.
%H A327932 Antti Karttunen, <a href="/A327932/b327932.txt">Table of n, a(n) for n = 1..65537</a>
%F A327932 a(n) = A327928(n) - A129251(n).
%F A327932 For n > 1, a(n) = A129251(A003415(n)) - A129251(n).
%e A327932 For n = 15 = 3*5, A129251(15) = 0, but for A003415(15) = 8 = 2^3, A129251(8) = 1, thus a(15) = 1.
%e A327932 For n = 515 = 5*103, A129251(515) = 0, but for A003415(515) = 108 = 2^2 * 3^3, A129251(108) = 2, thus a(515) = 2.
%o A327932 (PARI)
%o A327932 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 A327932 A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
%o A327932 A327928(n) = if(n<=1,0,A129251(A003415(n)));
%o A327932 A327932(n) = (A327928(n)-A129251(n));
%Y A327932 Cf. A003415, A129251, A327928, A327934.
%K A327932 nonn
%O A327932 1,515
%A A327932 _Antti Karttunen_, Oct 01 2019