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%I A374211 #8 Jun 30 2024 22:13:04
%S A374211 1,2,3,4,5,6,5,7,7,8,5,9,5,10,11,12,5,13,5,14,15,16,5,17,7,8,18,19,5,
%T A374211 16,5,20,21,22,23,24,5,25,26,27,5,28,5,29,30,31,5,32,7,33,17,34,5,35,
%U A374211 36,37,38,39,5,40,5,10,41,23,42,43,5,29,44,45,5,46,5,47,48,49,50,51,5,52,53,54,5,44,55,16,34,56,5,57,58,26,15,59,60,20,5,61,62,29
%N A374211 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), with f(1) = 1, and for n > 1, f(n) = [A278226(A328768(n)), A374212(n), A374213(n)], where A328768 is the first primorial based variant of the arithmetic derivative, and A374212 and A374213 are its 2- and 3-adic valuations.
%C A374211 Restricted growth sequence transform of the function f given in the definition.
%C A374211 For all i, j >= 1:
%C A374211   A305900(i) = A305900(j) => a(i) = a(j),
%C A374211   a(i) = a(j) => A152822(i) = A152822(j),
%C A374211   a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j),
%C A374211   a(i) = a(j) => A373991(i) = A373991(j).
%H A374211 Antti Karttunen, <a href="/A374211/b374211.txt">Table of n, a(n) for n = 1..100000</a>
%H A374211 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%o A374211 (PARI)
%o A374211 up_to = 100000;
%o A374211 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 A374211 A002110(n) = prod(i=1,n,prime(i));
%o A374211 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A374211 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A374211 A278226(n) = A046523(A276086(n));
%o A374211 A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
%o A374211 Aux374211(n) = if(1==n, n, my(u=A328768(n)); [A278226(u), valuation(u, 2), valuation(u, 3)]);
%o A374211 v374211 = rgs_transform(vector(up_to, n, Aux374211(n)));
%o A374211 A374211(n) = v374211[n];
%Y A374211 Cf. A002110, A278226, A328768, A374212, A374213.
%Y A374211 Cf. A305900, A152822, A328771, A373982, A373991.
%Y A374211 Cf. also A374131.
%K A374211 nonn
%O A374211 1,2
%A A374211 _Antti Karttunen_, Jun 30 2024

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