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A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.

%I A351036 #18 Jan 31 2022 18:10:41
%S A351036 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 A351036 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 A351036 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 A351036 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j) and A336158(i) = A336158(j), for all i, j >= 1.
%C A351036 Restricted growth sequence transform of the ordered pair [A000593(n), A336158(n)], where A000593(n) is the sum of odd divisors of n, and A336158(n) is the least representative of the prime signature of the odd part of n.
%C A351036 For all i, j:
%C A351036   A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
%C A351036   a(i) = a(j) => A113415(i) = A113415(j).
%H A351036 Antti Karttunen, <a href="/A351036/b351036.txt">Table of n, a(n) for n = 1..65537</a>
%o A351036 (PARI)
%o A351036 up_to = 65537;
%o A351036 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 A351036 A000265(n) = (n>>valuation(n,2));
%o A351036 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A351036 A336158(n) = A046523(A000265(n));
%o A351036 A000593(n) = sigma(A000265(n));
%o A351036 Aux351036(n) = [A000593(n), A336158(n)];
%o A351036 v351036 = rgs_transform(vector(up_to, n, Aux351036(n)));
%o A351036 A351036(n) = v351036[n];
%Y A351036 Cf. A000265, A000593, A003602, A046523, A113415, A336158.
%Y A351036 Cf. also A351037.
%Y A351036 Differs from A347374 for the first time at n=103, where a(103) = 48, while A347374(103) = 30.
%Y A351036 Differs from A351035 for the first time at n=175, where a(175) = 80, while A351035(175) = 78.
%Y A351036 Differs from A351040 for the first time at n=637, where a(637) = 261, while A351040(637) = 272.
%K A351036 nonn,easy
%O A351036 1,3
%A A351036 _Antti Karttunen_, Jan 30 2022