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%I #15 Jul 23 2022 09:55:35
%S 1,2,2,3,2,4,2,5,3,6,2,7,2,8,4,9,2,10,2,11,6,12,2,13,3,14,5,15,2,16,2,
%T 17,18,19,20,21,2,22,12,23,2,24,2,25,26,27,2,28,3,29,30,31,2,32,6,33,
%U 19,34,2,35,2,36,11,37,8,38,2,39,40,41,2,42,2,43,44,45,20,46,2,47,9,48,2,49,12,50,51,52,2,53,54,55,34,56,57,58,2,59,60,61,2,62,2,63,16
%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.
%C Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).
%H Antti Karttunen, <a href="/A355834/b355834.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
%e a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
%e a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.
%o (PARI)
%o up_to = 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 A000265(n) = (n>>valuation(n,2));
%o A009194(n) = gcd(n, sigma(n));
%o A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
%o Aux355834(n) = [A000265(A009194(n)), A348717(n)];
%o v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
%o A355834(n) = v355834[n];
%Y Cf. A000203, A000265, A009194, A246278, A348717, A355931.
%Y Cf. also A300230, A305800, A305897, A355833.
%K nonn
%O 1,2
%A _Antti Karttunen_, Jul 20 2022
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