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A351035 Lexicographically earliest infinite sequence such that a(i) = a(j) => A347385(i) = A347385(j) and A336158(i) = A336158(j), for all i, j >= 1.

%I A351035 #13 Jan 31 2022 18:10:36
%S A351035 1,1,2,1,3,2,4,1,5,3,6,2,7,4,8,1,9,5,10,3,11,6,12,2,13,7,14,4,15,8,16,
%T A351035 1,17,9,17,5,18,10,19,3,20,11,21,6,22,12,23,2,24,13,25,7,26,14,25,4,
%U A351035 27,15,28,8,29,16,30,1,31,17,32,9,33,17,34,5,35,18,36,10,33,19,37,3,38,20,39,11,40,21,41,6,42
%N A351035 Lexicographically earliest infinite sequence such that a(i) = a(j) => A347385(i) = A347385(j) and A336158(i) = A336158(j), for all i, j >= 1.
%C A351035 Restricted growth sequence transform of the ordered pair [A347385(n), A336158(n)], where A347385(n) is the Dedekind psi function applied to the odd part of n, i.e., A001615(A000265(n)), and A336158(n) is the least representative of the prime signature of the odd part of n.
%C A351035 For all i, j >= 1: A003602(i) = A003602(j) => a(i) = a(j).
%H A351035 Antti Karttunen, <a href="/A351035/b351035.txt">Table of n, a(n) for n = 1..65537</a>
%e A351035 a(33) = a(35) as both 33 = 3*11 and 35 = 5*7 are odd nonsquare semiprimes, thus A336158 gives equal values for them, and also A347385(33) = A001615(33) = A347385(35) = A001615(35) = 48.
%o A351035 (PARI)
%o A351035 up_to = 65537;
%o A351035 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 A351035 A000265(n) = (n>>valuation(n,2));
%o A351035 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A351035 A336158(n) = A046523(A000265(n));
%o A351035 A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
%o A351035 Aux351035(n) = [A347385(n), A336158(n)];
%o A351035 v351035 = rgs_transform(vector(up_to, n, Aux351035(n)));
%o A351035 A351035(n) = v351035[n];
%Y A351035 Cf. A000265, A001615, A003602, A046523, A336158, A351035.
%Y A351035 Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
%Y A351035 Differs from A351036 for the first time at n=175, where a(175) = 78, while A351036(175) = 80.
%K A351035 nonn,easy
%O A351035 1,3
%A A351035 _Antti Karttunen_, Jan 30 2022