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A379240 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j, where f(n) = [A003415(n), A085731(n)] if A359550(n) = 1, otherwise f(n) = n.

%I A379240 #8 Dec 19 2024 21:15:42
%S A379240 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,
%T A379240 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,
%U A379240 28,40,41,21,2,42,2,43,44,45,46,47,2,48,49,50,2,51,2,52,53,54,46,55,2,56,57,58,2,59,41,60,61,62,2,63,37,64
%N A379240 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j, where f(n) = [A003415(n), A085731(n)] if A359550(n) = 1, otherwise f(n) = n.
%C A379240 It is conjectured that this is also the lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A376418(i) = A376418(j), for all i, j >= 1, i.e., the restricted growth sequence transform of the triple [A003415(n), A085731(n), A376418(n)]. This is true if for every pair of i and j for which i <> j, and A376418(i) = A376418(j) > 0, the ordered pairs [A003415(i), A085731(i)] and [A003415(j), A085731(j)] differ from each other.
%H A379240 Antti Karttunen, <a href="/A379240/b379240.txt">Table of n, a(n) for n = 1..100000</a>
%F A379240 For all i, j >= 1:
%F A379240   a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),
%F A379240   a(i) = a(j) => A376418(i) = A376418(j).
%o A379240 (PARI)
%o A379240 up_to = 100000;
%o A379240 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 A379240 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A379240 A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
%o A379240 Aux379240(n) = if(!A359550(n), n, my(d=A003415(n)); [d, gcd(d,n)]);
%o A379240 v379240 = rgs_transform(vector(up_to, n, Aux379240(n)));
%o A379240 A379240(n) = v379240[n];
%Y A379240 Cf. A003415, A048103, A083345, A085371, A100716, A359550, A369051, A376418.
%Y A379240 Differs from A344025 first at n=140, where a(140) = 97, while A344025(140) = 92.
%Y A379240 Differs from A369046 first at n=171, where a(171) = 63, while A369046(171) = 121.
%K A379240 nonn
%O A379240 1,2
%A A379240 _Antti Karttunen_, Dec 19 2024