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A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

%I A355835 #17 Jul 28 2022 09:17:52
%S A355835 1,2,3,4,3,5,3,6,7,8,3,9,3,10,5,11,3,12,3,13,14,15,3,16,17,18,19,20,3,
%T A355835 21,3,22,23,24,25,26,3,27,28,29,3,30,3,31,32,33,3,34,17,35,36,37,3,38,
%U A355835 39,40,41,42,3,43,3,44,45,46,47,48,3,49,50,51,3,52,3,53,54,55,56,57,3,58,59,60,3,61,62,63,64,65,3,66,67,68,69,70,71,72,3,73,74
%N A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.
%C A355835 Restricted growth sequence transform of the ordered pair [A348717(n), A355442(n)].
%C A355835 For all i, j: a(i) = a(j) => A355836(i) = A355836(j).
%C A355835 Terms that occur in positions given by A355822 may occur only a finite number of times in this sequence. Most of these seem to be in the singular equivalence classes, i.e., have unique values, apart from exceptions like pairs {6, 15}, {273, 1729}, (see the examples and the array A355926). In a coarser variant A355836 multiple such finite equivalence classes may coalesce together into several infinite equivalence classes.
%H A355835 Antti Karttunen, <a href="/A355835/b355835.txt">Table of n, a(n) for n = 1..100000</a>
%H A355835 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A355835 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A355835 a(6) = a(15) [= 5 as allotted by the rgs-transform] because 15 = A003961(6) [i.e., 15 is in the same column in prime shift array A246278 as 6 is], and because A355442(6) = A355442(15) = 5.
%e A355835 a(138) = a(435) [= 103 as allotted by the rgs-transform] because 435 = A003961(138), and A355442(138) = A355442(435) = 5.
%e A355835 a(273) = a(1729) [= 205 as allotted by the rgs-transform] because 1729 = A003961(A003961(273)) [i.e., 273 and 1729 are in the same column of A246278], and A355442(273) = A355442(1729) = 11.
%o A355835 (PARI)
%o A355835 up_to = 65537;
%o A355835 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 A355835 A348717(n) = if(1==n, 1, 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));
%o A355835 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A355835 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A355835 A355442(n) = gcd(A003961(n), A276086(n));
%o A355835 Aux355835(n) = [A348717(n), A355442(n)];
%o A355835 v355835 = rgs_transform(vector(up_to,n,Aux355835(n)));
%o A355835 A355835(n) = v355835[n];
%Y A355835 Cf. A003961, A276086, A348717, A355442, A355821, A355822, A355926.
%Y A355835 Cf. also A246278, A305801, A355833, A355834.
%K A355835 nonn
%O A355835 1,2
%A A355835 _Antti Karttunen_, Jul 20 2022