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A300246 Filter sequence combining A046523(n) and A078899(n), the prime signature of n and the number of times the greatest prime factor of n is the greatest prime factor for numbers <= n.

%I A300246 #6 Mar 09 2018 17:52:19
%S A300246 1,2,2,3,2,4,2,5,6,4,2,7,2,4,8,9,2,10,2,7,8,4,2,11,12,4,13,7,2,14,2,
%T A300246 15,8,4,16,17,2,4,8,18,2,14,2,7,19,4,2,20,21,22,8,7,2,23,16,24,8,4,2,
%U A300246 25,2,4,22,26,16,14,2,7,8,27,2,28,2,4,29,7,30,14,2,31,32,4,2,33,16,4,8,24,2,34,30,7,8,4,16,35,2,36,22,37,2,14,2,24,38
%N A300246 Filter sequence combining A046523(n) and A078899(n), the prime signature of n and the number of times the greatest prime factor of n is the greatest prime factor for numbers <= n.
%C A300246 Restricted growth sequence transform of P(A046523(n), A078899(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
%H A300246 Antti Karttunen, <a href="/A300246/b300246.txt">Table of n, a(n) for n = 1..65537</a>
%e A300246 a(30) = a(42) (= 14) because A078899(30) = A078899(42) = 6 and both numbers are products of three distinct primes, thus have the same prime signature.
%e A300246 a(35) = a(55) = a(65) (= 16) because A078899(35) = A078899(55) = A078899(65) = 5 and because all three are nonsquare semiprimes.
%o A300246 (PARI)
%o A300246 up_to = 65537;
%o A300246 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 A300246 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A300246 A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
%o A300246 A078899(n) = { if(n<=1,n, my(gpf=A006530(n),k=1,m=n/gpf); while(m>1,if(A006530(m)<=gpf,k++); m--); (k)); };
%o A300246 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A300246 Aux300246(n) = if(1==n,0,(1/2)*(2 + ((A078899(n)+A046523(n))^2) - A078899(n) - 3*A046523(n)));
%o A300246 write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300246(n))),"b300246.txt");
%Y A300246 Cf. A046523, A078899.
%Y A300246 Cf. also A300247, A300248.
%Y A300246 Differs from A300226 for the first time at n=40, where a(40) = 18.
%K A300246 nonn
%O A300246 1,2
%A A300246 _Antti Karttunen_, Mar 09 2018