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A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= n.

%I A300248 #8 Mar 03 2018 17:34:53
%S A300248 1,2,2,3,2,4,2,5,3,6,2,7,2,8,4,9,2,10,2,11,12,13,2,14,3,15,16,17,2,18,
%T A300248 2,19,20,21,4,22,2,23,8,24,2,25,2,26,27,28,2,29,3,30,31,32,2,33,12,34,
%U A300248 35,36,2,37,2,38,39,40,6,41,2,42,43,44,2,45,2,46,47,48,4,49,2,50,51,52,2,53,20,54,55,56,2,57,12,58,59,60,8,61,2,62,63,64,2
%N A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= n.
%C A300248 Restricted growth sequence transform of P(A046523(n), A078898(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
%H A300248 Antti Karttunen, <a href="/A300248/b300248.txt">Table of n, a(n) for n = 1..65537</a>
%e A300248 a(10) = a(65) (= 6) because A078898(10) = A078898(65) = 5 (both numbers occur in column 5 of A083221) and because both have the same prime-signature (both are nonsquare semiprimes).
%o A300248 (PARI)
%o A300248 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 A300248 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A300248 A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
%o A300248 A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
%o A300248 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A300248 A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
%o A300248 Aux300248(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A046523(n))^2) - A078898(n) - 3*A046523(n)));
%o A300248 write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300248.txt");
%Y A300248 Cf. A046523, A078898.
%Y A300248 Cf. also A300226, A300229, A300247.
%K A300248 nonn
%O A300248 1,2
%A A300248 _Antti Karttunen_, Mar 03 2018