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A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).

%I A319716 #15 Oct 05 2018 11:10:42
%S A319716 1,2,3,4,5,6,7,8,9,10,5,11,7,12,13,14,5,15,7,16,17,18,5,19,20,21,22,
%T A319716 23,5,24,7,25,26,27,28,29,7,30,31,32,5,33,7,34,35,36,5,37,38,39,40,41,
%U A319716 5,42,43,44,45,46,5,47,7,48,49,50,51,52,7,53,54,55,5,56,7,57,58,59,60,61,7,62,63,64,5,65,66,67,68,69,5,70,71,72,73,74,75,76,7,77,78,79,5,80,7,81,82,83,5,84,7,85,86,87,5,88,89,90,91,92,93,94,43
%N A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).
%C A319716 Restricted growth sequence transform of A286475, or equally, of A286476.
%C A319716 In each a(n) there is enough information to determine the modulo 6 residues of all the prime factors of n (when counted with multiplicity), thus sequences like A319690 and A319691 (which is the characteristic function of A004611) are essentially functions of this sequence. However, to determine that for all divisors of n, more information is needed. See A319717.
%C A319716 For all i, j:
%C A319716   A319707(i) = A319707(j) => A319717(i) = A319717(j) => a(i) = a(j),
%C A319716   a(i) = a(j) => A319690(i) = A319690(i) => A319691(i) = A319691(j).
%H A319716 Antti Karttunen, <a href="/A319716/b319716.txt">Table of n, a(n) for n = 1..100000</a>
%e A319716 For n = 55 = 5*11 and 121 = 11*11, 55 = 121 = 1 mod 6 and 11 is their common largest proper divisor, thus they are allotted the same number by the restricted growth sequence transform, that is a(55) = a(121) = 43 (which is the number allotted). Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3.
%o A319716 (PARI)
%o A319716 up_to = 100000;
%o A319716 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 A319716 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A319716 A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
%o A319716 v319716 = rgs_transform(vector(up_to,n,A286476(n)));
%o A319716 A319716(n) = v319716[n];
%Y A319716 Cf. A010875, A032742, A286475, A286476, A319707, A319717.
%Y A319716 Cf. A007528 (positions of 5's), A002476 (positions of 7's).
%Y A319716 Cf. also A319714.
%Y A319716 Differs from A319707 and A319717 for the first time at n=121.
%K A319716 nonn
%O A319716 1,2
%A A319716 _Antti Karttunen_, Oct 04 2018