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A372575 Lexicographically earliest infinite sequence such that a(i) = a(j) => A276085(i) = A276085(j), for all i, j >= 1, where A276085 is the primorial base log-function.

%I A372575 #17 May 26 2024 08:29:53
%S A372575 1,2,3,3,4,5,6,5,7,8,9,7,10,11,12,7,13,14,15,12,16,17,18,14,19,20,4,
%T A372575 16,21,22,23,14,24,25,26,4,27,28,29,22,30,31,32,24,33,34,35,4,36,37,
%U A372575 38,29,39,8,40,31,41,42,43,33,44,45,46,4,47,48,49,38,50,51,52,8,53,54,55,41,56,57,58,33,12,59,60,46,61,62,63,48,64,65,66,50,67,68,69,8
%N A372575 Lexicographically earliest infinite sequence such that a(i) = a(j) => A276085(i) = A276085(j), for all i, j >= 1, where A276085 is the primorial base log-function.
%C A372575 Restricted growth sequence transform of A276085, where A276085 is fully additive with a(p) = A002110(A000720(p)-1).
%C A372575 For all i, j >= 1:
%C A372575   a(i) = a(j) => A035263(i) = A035263(j),
%C A372575   a(i) = a(j) => A369001(i) = A369001(j),
%C A372575   a(i) = a(j) => A372576(i) = A372576(j).
%C A372575 A048103(n) gives the position of the initial occurrence of each n, while A100716 gives the indices of the further occurrences.
%H A372575 Antti Karttunen, <a href="/A372575/b372575.txt">Table of n, a(n) for n = 1..65537</a>
%H A372575 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%H A372575 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A372575 For all n >= 1, a(A048103(n)) = n.
%o A372575 (PARI)
%o A372575 up_to = 65537;
%o A372575 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 A372575 A002110(n) = prod(i=1,n,prime(i));
%o A372575 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
%o A372575 v372575 = rgs_transform(vector(up_to, n, A276085(n)));
%o A372575 A372575(n) = v372575[n];
%Y A372575 Cf. A000720, A002110, A276085, A035263, A369001, A372576.
%Y A372575 Cf. A048103 (positions of records and initial occurrences), A100716.
%Y A372575 Cf. also A322815.
%K A372575 nonn
%O A372575 1,2
%A A372575 _Antti Karttunen_, May 25 2024