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A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

%I A319685 #13 Apr 17 2024 02:30:51
%S A319685 0,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2,
%T A319685 2,7,1,2,2,6,1,5,1,4,4,2,1,8,2,4,2,4,1,6,2,6,2,2,1,9,1,2,4,6,2,5,1,4,
%U A319685 2,5,1,10,1,2,4,4,2,5,1,8,4,2,1,9,2,2,2,6,1,9,2,4,2,2,2,10,1,4,4,7,1,5,1,6,5,2,1,10,1,5,2,7,1,5,2,4,4,2,2,13
%N A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.
%C A319685 Because for all d|n, d<n, A003415(d) < A003415(n), it follows that the terms here are one less than in A319686.
%C A319685 Differs from A033273(n) = A000005(n) - A001221(n) at n = 1, 112, 156, 224, 280, 312, 336, 342, 380, 448, 468, 525, 560, 608, 624, 660, 672, 684, 756, 760, 780, 784, 840, 870, 896, 936, 984, 1008, ...
%H A319685 Antti Karttunen, <a href="/A319685/b319685.txt">Table of n, a(n) for n = 1..65537</a>
%F A319685 a(n) = A319686(n)-1.
%e A319685 The proper divisors of 112 are [1, 2, 4, 7, 8, 14, 16, 28, 56]. Applying arithmetic derivative A003415 to these, we obtain values [0, 1, 4, 1, 12, 9, 32, 32, 92], of which only 7 are distinct: 0, 1, 4, 9, 12, 32, and 92, thus a(112) = 7.
%t A319685 d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Most[Divisors[n]]]; Array[a, 100] (* _Amiram Eldar_, Apr 17 2024 *)
%o A319685 (PARI)
%o A319685 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 A319685 A319685(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d<n)&&!mapisdefined(m,s=A003415(d)), mapput(m,s,s); k++)); (k); };
%Y A319685 One less than A319686.
%Y A319685 Cf. A003415.
%Y A319685 Cf. also A304793, A305611, A316555, A316556, A319695 for similarly constructed sequences.
%K A319685 nonn
%O A319685 1,4
%A A319685 _Antti Karttunen_, Oct 02 2018