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A355836 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A355442(i) = A355442(j) for all i, j >= 1.

%I A355836 #16 Jul 28 2022 09:18:37
%S A355836 1,2,3,4,3,5,3,6,7,8,3,9,3,8,5,10,3,11,3,12,5,8,3,13,14,8,15,16,3,17,
%T A355836 3,18,19,8,20,21,3,8,5,22,3,23,3,12,24,8,3,25,14,26,5,16,3,27,20,28,5,
%U A355836 8,3,29,3,8,30,31,20,23,3,12,5,32,3,33,3,8,34,16,19,23,3,35,36,8,3,37,20,8,5,38,3,29,19,12,19,8,20,39,3,12,9,40,3,23,3,28,41
%N A355836 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A355442(i) = A355442(j) for all i, j >= 1.
%C A355836 Restricted growth sequence transform of the ordered pair [A046523(n), A355442(n)].
%C A355836 For all i, j: A355835(i) = A355835(j) => a(i) = a(j).
%H A355836 Antti Karttunen, <a href="/A355836/b355836.txt">Table of n, a(n) for n = 1..100000</a>
%H A355836 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A355836 a(6) = a(15) = a(21) = a(39) = a(51) = a(57) = a(69) = a(87) = a(111) = etc, for an infinite number of other indices k, because for all these k, A355442(k) = 5 and their prime signatures (A101296) are equal, as they are all squarefree semiprimes, A006881.
%e A355836 In contrast, powers of 2 (1, 2, 4, 8, 16, ..., A000079) obtain unique values in this sequence, and in general, for all proper prime powers k (A246547) for which A355442(k) > 1 [that are terms of A355822], the value a(k) is unique in this sequence.
%o A355836 (PARI)
%o A355836 up_to = 100000;
%o A355836 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 A355836 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A355836 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A355836 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A355836 A355442(n) = gcd(A003961(n), A276086(n));
%o A355836 Aux355836(n) = [A046523(n), A355442(n)];
%o A355836 v355836 = rgs_transform(vector(up_to,n,Aux355836(n)));
%o A355836 A355836(n) = v355836[n];
%Y A355836 Cf. A000079, A003961, A006881, A046523, A101296, A246547, A276086, A355442, A355822, A355835.
%Y A355836 Cf. also A355000, A355830.
%K A355836 nonn
%O A355836 1,2
%A A355836 _Antti Karttunen_, Jul 20 2022