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A374032 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A374031(i)) = A278226(A374031(j)), for all i, j >= 1.

%I A374032 #8 Jun 27 2024 10:20:38
%S A374032 1,2,2,2,2,2,2,3,4,2,2,4,2,2,4,4,2,2,2,4,3,2,2,2,4,2,2,4,2,2,2,5,4,2,
%T A374032 4,2,2,2,4,6,2,2,2,4,2,2,2,2,4,2,4,4,2,2,3,3,7,2,2,2,2,2,2,2,4,2,2,4,
%U A374032 4,2,2,2,2,2,2,4,8,2,2,2,3,2,2,2,4,2,4,3,2,2,9,4,3,2,4,2,2,2,2,2,2,2,2,3,6
%N A374032 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A374031(i)) = A278226(A374031(j)), for all i, j >= 1.
%C A374032 Restricted growth sequence transform of A278226(A374031(n)).
%C A374032 For all i, j >= 1:
%C A374032   A305800(i) = A305800(j) => a(i) = a(j) => A374034(i) = A374034(j).
%H A374032 Antti Karttunen, <a href="/A374032/b374032.txt">Table of n, a(n) for n = 1..100000</a>
%H A374032 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%o A374032 (PARI)
%o A374032 up_to = 100000;
%o A374032 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 A374032 A002110(n) = prod(i=1,n,prime(i));
%o A374032 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
%o A374032 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
%o A374032 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A374032 A278226(n) = A046523(A276086(n));
%o A374032 A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
%o A374032 A374031(n) = gcd(A276085(n), A328768(n));
%o A374032 v374032 = rgs_transform(vector(up_to, n, A278226(A374031(n))));
%o A374032 A374032(n) = v374032[n];
%Y A374032 Cf. A002110, A046523, A276085, A276086, A278226, A328768, A374031.
%Y A374032 Cf. A305800, A374034.
%Y A374032 Cf. also A373982, A374033.
%K A374032 nonn
%O A374032 1,2
%A A374032 _Antti Karttunen_, Jun 27 2024