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A305897 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j), for all i, j >= 1.

%I A305897 #20 Apr 30 2022 22:52:03
%S A305897 1,2,2,3,2,4,2,5,3,6,2,7,2,8,4,9,2,10,2,11,6,12,2,13,3,14,5,15,2,16,2,
%T A305897 17,8,18,4,19,2,20,12,21,2,22,2,23,7,24,2,25,3,26,14,27,2,28,6,29,18,
%U A305897 30,2,31,2,32,11,33,8,34,2,35,20,36,2,37,2,38,10,39,4,40,2,41,9,42,2,43,12,44,24,45,2,46,6,47,30,48,14,49,2,50,15,51,2,52,2,53,16
%N A305897 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j), for all i, j >= 1.
%C A305897 Restricted growth sequence transform of A348717, or equally, of A246277.
%C A305897 Filter sequence for prime factorization patterns, including also information about gaps between prime factors. - Original name, gives the motivation for this sequence. Here the "gaps" refers to differences between the indices of primes present, not the prime gaps as usually understood.
%C A305897 For all i, j:
%C A305897   A305800(i) = A305800(j) => a(i) = a(j),
%C A305897   a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).
%C A305897   a(i) = a(j) => A243055(i) = A243055(j).
%H A305897 Antti Karttunen, <a href="/A305897/b305897.txt">Table of n, a(n) for n = 1..100000</a>
%H A305897 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A305897 a(10) = a(21) (= 6) because both have prime exponents [1, 1] and the difference between the prime indices is the same, as 10 = prime(1)*prime(3), and 21 = prime(2)*prime(4).
%e A305897 a(12) != a(18) because the prime exponents [2,1] and [1,2] do not occur in the same order.
%e A305897 a(140) = a(693) (= 71) because both numbers have prime exponents [2, 1, 1] (in this order) and the differences between the indices of the successive prime factors are same: 140 = prime(1)^2 * prime(3) * prime(4), 693 = prime(2)^2 * prime(4) * prime(5).
%o A305897 (PARI)
%o A305897 up_to = 65537;
%o A305897 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 A305897 A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
%o A305897 v305897 = rgs_transform(vector(up_to,n,A246277(n)));
%o A305897 A305897(n) = v305897[n];
%Y A305897 Cf. A077462, A101296, A246277, A300247, A305800, A348717.
%Y A305897 Cf. also A286621.
%K A305897 nonn
%O A305897 1,2
%A A305897 _Antti Karttunen_, Jun 14 2018
%E A305897 Name changed by _Antti Karttunen_, Apr 30 2022