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%I #7 Mar 04 2018 11:49:10
%S 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,
%T 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,
%U 28,40,41,42,2,43,2,44,45,46,47,48,2,49,50,51,2,52,2,53,54,55,47,56,2,57,58,59,2,60,41,61,62,63,2,64,37,65,66,67
%N Filter sequence combining A003415(n) and A046523(n), the arithmetic derivative of n and the prime signature of n.
%C Restricted growth sequence transform of P(A003415(n), A046523(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
%H Antti Karttunen, <a href="/A300249/b300249.txt">Table of n, a(n) for n = 1..65537</a>
%e a(51) = a(91) (= 33) because both are nonsquare semiprimes (3*17 and 7*13), and also their arithmetic derivatives are equal, as A003415(51) = A003415(91) = 20.
%e a(78) = a(105) (= 56) because both have the same prime signature (78 = 2*3*13 and 105 = 3*5*7), and also their arithmetic derivatives are equal, as A003415(78) = A003415(105) = 71.
%o (PARI)
%o rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o 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 A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
%o Aux300249(n) = ((1/2)*(2 + ((A003415(n)+A046523(n))^2) - A003415(n) - 3*A046523(n)));
%o write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300249.txt");
%Y Cf. A003415, A046523.
%Y Cf. also A300226, A300229, A300248.
%Y Differs from A300235 for the first time at n=105, where a(105)=56, while A300235(105)=75.
%K nonn
%O 1,2
%A _Antti Karttunen_, Mar 04 2018
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