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A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(j), for all i, j >= 1.

%I A351090 #15 Feb 01 2022 18:57:21
%S A351090 1,1,2,1,3,2,4,1,5,3,6,2,7,4,8,1,9,5,10,3,11,6,12,2,13,7,14,4,15,8,16,
%T A351090 1,17,9,18,5,19,10,20,3,21,11,22,6,23,12,24,2,25,13,26,7,27,14,28,4,
%U A351090 29,15,30,8,31,16,32,1,33,17,34,9,35,18,36,5,37,19,38,10,39,20,40,3,41,21,42,11,43,22,44,6,45,23
%N A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(j), for all i, j >= 1.
%C A351090 Restricted growth sequence transform of the ordered pair [A351091(n), A351092(n)], or equally, of the ordered pair [A351093(n), A351094(n)].
%C A351090 For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A000593(i) = A000593(j).
%H A351090 Antti Karttunen, <a href="/A351090/b351090.txt">Table of n, a(n) for n = 1..65537</a>
%e A351090 Consider two odd semiprimes, 689 and 697. The divisors of 689 are 1, 13, 53, 689, and the divisors of 697 are 1, 17, 41, 697. Applying A019565(A289813(x)) to the former gives [2, 30, 7, 105], while with the latter it gives [2, 5, 105, 42], and the product of both sequences is 44100. Applying A019565(A289814(x)) to the former gives [1, 1, 30, 286], while with the latter it gives [1, 6, 2, 715]. Product of both sequences is 8580. Therefore, because A351091(689) = A351091(697) and A351092(689) = A351092(697), also a(689) = a(697).
%o A351090 (PARI)
%o A351090 up_to = 65537;
%o A351090 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 A351090 A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
%o A351090 A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); }; \\ From A289813
%o A351090 A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); }; \\ From A289814
%o A351090 A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };
%o A351090 A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };
%o A351090 Aux351090(n) = [A351091(n),A351092(n)];
%o A351090 v351090 = rgs_transform(vector(up_to, n, Aux351090(n)));
%o A351090 A351090(n) = v351090[n];
%Y A351090 Cf. A000593, A351037, A351091, A351092, A351093, A351094.
%Y A351090 Differs from A003602 for the first time at n=697, where a(697) = 345 while A003602(697) = 349.
%Y A351090 Cf. also A293226, A351030.
%K A351090 nonn,easy
%O A351090 1,3
%A A351090 _Antti Karttunen_, Jan 31 2022