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A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

%I A319686 #15 Apr 17 2024 02:30:32
%S A319686 1,2,2,3,2,3,2,4,3,3,2,5,2,3,3,5,2,5,2,5,3,3,2,7,3,3,4,5,2,6,2,6,3,3,
%T A319686 3,8,2,3,3,7,2,6,2,5,5,3,2,9,3,5,3,5,2,7,3,7,3,3,2,10,2,3,5,7,3,6,2,5,
%U A319686 3,6,2,11,2,3,5,5,3,6,2,9,5,3,2,10,3,3,3,7,2,10,3,5,3,3,3,11,2,5,5,8,2,6,2,7,6,3,2,11,2,6,3,8,2,6,3,5,5,3,3,14
%N A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.
%H A319686 Antti Karttunen, <a href="/A319686/b319686.txt">Table of n, a(n) for n = 1..65537</a>
%F A319686 a(n) = 1+A319685(n).
%t A319686 d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Divisors[n]]; Array[a, 100] (* _Amiram Eldar_, Apr 17 2024 *)
%o A319686 (PARI)
%o A319686 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 A319686 A319686(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A003415(d)), mapput(m,s,s); k++)); (k); };
%o A319686 (PARI) a(n) = my(d = divisors(n)); for(i = 1, #d, d[i] = A003415(d[i])); #Set(d) \\ uses A003415 listed at Antti's programs. _David A. Corneth_, Oct 02 2018
%Y A319686 One more than A319685.
%Y A319686 Cf. A003415.
%Y A319686 Cf. also A184395, A319696.
%K A319686 nonn
%O A319686 1,2
%A A319686 _Antti Karttunen_, Oct 02 2018