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%I #8 Jul 17 2020 22:58:39
%S 1,2,2,2,2,2,2,3,3,2,2,3,2,2,3,3,2,2,2,3,3,2,2,3,2,2,4,3,2,2,2,3,3,2,
%T 2,5,2,2,3,3,2,2,2,3,4,2,2,3,3,3,3,3,2,2,2,3,3,2,2,5,2,2,4,5,2,2,2,3,
%U 3,3,2,5,2,2,3,3,3,2,2,3,6,2,2,5,2,2,3,3,2,2,3,3,3,2,2,5,2,2,4,3,2,2,2,3,3
%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A336125(i)) = A278222(A336125(j)) for all i, j >= 1.
%C Restricted growth sequence transform of A278222(A336125(n)).
%C For all i, j: A336311(i) = A336311(j) => a(i) = a(j) => A336123(i) = A336123(j).
%H Antti Karttunen, <a href="/A336313/b336313.txt">Table of n, a(n) for n = 1..65537</a>
%o (PARI)
%o \\ Needs also code from A336124:
%o up_to = 1024; \\ 65537;
%o 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 A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
%o A336125(n) = if(n<=2,n-1,(1==A336124(n))+(2*A336125(A253553(n))));
%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o A278222(n) = A046523(A005940(1+n));
%o v336313 = rgs_transform(vector(up_to,n,A278222(A336125(n))));
%o A336313(n) = v336313[n];
%Y Cf. A253553, A278222, A336124, A336125.
%Y Cf. also A292585, A336311, A336312, A336123.
%K nonn
%O 1,2
%A _Antti Karttunen_, Jul 17 2020
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