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A327931 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).

%I A327931 #20 Sep 30 2019 20:20:17
%S A327931 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 A327931 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 A327931 28,40,41,42,2,43,2,44,45,46,47,48,2,49,50,51,2,52,2,53,54,55,47,56,2,57,58,59,2,60,41,61,62,63,2,64,37,65,66,67,68,69,2,70,71,72,2,73,2,74,75
%N A327931 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).
%C A327931 Restricted growth sequence transform of A327930, or equally, of the ordered pair [A003415(n), A319356(n)].
%C A327931 It seems that the sequence takes duplicated values only on primes (A000040) and some subset of squarefree semiprimes (A006881). If this holds, then also the last implication given below is valid.
%C A327931 For all i, j:
%C A327931   a(i) = a(j) => A000005(i) = A000005(j),
%C A327931   a(i) = a(j) => A319684(i) = A319684(j),
%C A327931   a(i) = a(j) => A319685(i) = A319685(j),
%C A327931   a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above and in A319357]
%H A327931 Antti Karttunen, <a href="/A327931/b327931.txt">Table of n, a(n) for n = 1..65537</a>
%F A327931 a(p) = 2 for all primes p.
%e A327931 Divisors of 39 are [1, 3, 13, 39], while the divisors of 55 are [1, 5, 11, 55]. Taking their arithmetic derivatives (A003415) yields in both cases [0, 1, 1, 16], thus a(39) = a(55) (= 28, as allotted by restricted growth sequence transform).
%o A327931 (PARI)
%o A327931 up_to = 8192;
%o A327931 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 A327931 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A327931 v003415 = vector(up_to,n,A003415(n));
%o A327931 A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(v003415[d]))); (m); };
%o A327931 v327931 = rgs_transform(vector(up_to, n, A327930(n)));
%o A327931 A327931(n) = v327931[n];
%Y A327931 Cf. A000005, A000040 (positions of 2's), A003415, A006881, A101296, A319356, A319357, A319684, A319685, A327930.
%Y A327931 Differs from A300249 for the first time at n=105, where a(105)=75, while A300249(105)=56.
%Y A327931 Differs from A300235 for the first time at n=153, where a(153)=110, while A300235(153)=106.
%Y A327931 Differs from A305895 for the first time at n=3283, where a(3283)=2502, while A305895(3283)=1845.
%K A327931 nonn
%O A327931 1,2
%A A327931 _Antti Karttunen_, Sep 30 2019