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A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.

%I A373983 #12 Jun 26 2024 09:32:26
%S A373983 1,2,2,3,2,4,2,5,5,6,2,7,2,8,9,10,2,11,2,12,13,14,2,15,16,6,17,18,2,
%T A373983 19,2,20,21,22,23,24,2,25,26,27,2,28,2,29,30,31,2,32,33,34,13,35,2,36,
%U A373983 37,38,39,40,2,41,2,8,42,43,44,45,2,29,46,47,2,48,2,49,50,51,52,53,2,54,55,56,2,57,58,14,59,60,2,61,62,63,13,64,65,66,2,67,68,69
%N A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.
%C A373983 Restricted growth sequence transform of the ordered pair [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))].
%C A373983 For all i, j >= 1:
%C A373983   A305800(i) = A305800(j) => a(i) = a(j),
%C A373983   a(i) = a(j) => A329345(i) = A329345(j) => A329045(i) = A329045(j),
%C A373983   a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j).
%C A373983 It is hard to say for sure which graphical features in the scatter plot have their provenance in A373982, and which ones in A329345.
%H A373983 Antti Karttunen, <a href="/A373983/b373983.txt">Table of n, a(n) for n = 1..100000</a>
%H A373983 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%o A373983 (PARI)
%o A373983 up_to = 100000;
%o A373983 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 A373983 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
%o A373983 A108951(n) = { my(f=factor(n)); prod(i=1, #f~,  prod(i=1, primepi(f[i, 1]), prime(i))^f[i, 2]); };
%o A373983 A246277(n) = if(1==n, 0, 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)/2);
%o A373983 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A373983 A002110(n) = prod(i=1,n,prime(i));
%o A373983 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 A373983 Aux373983(n) = [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))];
%o A373983 v373983 = rgs_transform(vector(up_to, n, Aux373983(n)));
%o A373983 A373983(n) = v373983[n];
%Y A373983 Cf. A002110, A046523, A108951, A246277, A276086, A278226, A324886, A328768.
%Y A373983 Cf. A305800, A329045, A329345, A328771, A373982.
%Y A373983 Cf. also A329620.
%K A373983 nonn,look
%O A373983 1,2
%A A373983 _Antti Karttunen_, Jun 25 2024