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A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

%I A294877 #26 Dec 20 2018 23:17:01
%S A294877 1,2,2,3,2,4,2,5,6,4,2,7,2,4,4,8,2,9,2,7,4,4,2,10,11,4,12,7,2,13,2,14,
%T A294877 4,4,4,15,2,4,4,10,2,13,2,7,9,4,2,16,17,18,4,7,2,19,4,10,4,4,2,20,2,4,
%U A294877 9,21,4,13,2,7,4,13,2,22,2,4,18,7,4,13,2,16,23,4,2,20,4,4,4,10,2,24,4,7,4,4,4,25,2,26,9,27,2,13,2,10,13,4,2,28,2,13
%N A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.
%C A294877 Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
%C A294877 Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
%C A294877 For all i, j:
%C A294877   A295300(i) = A295300(j) => a(i) = a(j),
%C A294877   A319347(i) = A319347(j) => a(i) = a(j),
%C A294877   a(i) = a(j) => A055155(i) = A055155(j).
%H A294877 Antti Karttunen, <a href="/A294877/b294877.txt">Table of n, a(n) for n = 1..65537</a>
%o A294877 (PARI)
%o A294877 up_to = 65537;
%o A294877 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 A294877 A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
%o A294877 v294877 = rgs_transform(vector(up_to,n,A294876(n)));
%o A294877 A294877(n) = v294877[n];
%o A294877 (PARI)
%o A294877 A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
%o A294877 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A294877 v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
%o A294877 A294877(n) = v294877[n]; \\ _Antti Karttunen_, Nov 28 2018
%Y A294877 Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.
%Y A294877 Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).
%K A294877 nonn
%O A294877 1,2
%A A294877 _Antti Karttunen_, Nov 11 2017
%E A294877 Name changed and comments added by _Antti Karttunen_, Nov 28 2018