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A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

%I A326199 #6 Jul 13 2019 09:12:04
%S A326199 1,2,3,4,3,5,3,6,7,8,3,9,3,10,10,11,3,12,3,13,14,15,3,16,17,18,19,20,
%T A326199 3,21,3,22,23,24,23,25,3,26,27,28,3,29,3,30,31,32,3,33,34,35,32,36,3,
%U A326199 37,32,38,39,40,3,41,3,42,43,44,45,46,3,47,42,46,3,48,3,49,50,51,42,52,3,53,54,55,3,56,57,58,59,60,3,61,62,63,64,65,59,66,3,67,68,69,3,70,3
%N A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.
%C A326199 For all i, j:
%C A326199   A295300(i) = A295300(j) => a(i) = a(j),
%C A326199   A305801(i) = A305801(j) => a(i) = a(j),
%C A326199   a(i) = a(j) => A294877(i) = A294877(j).
%H A326199 Antti Karttunen, <a href="/A326199/b326199.txt">Table of n, a(n) for n = 1..65537</a>
%o A326199 (PARI)
%o A326199 up_to = 65537;
%o A326199 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 A326199 A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
%o A326199 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A326199 A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
%o A326199 A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
%o A326199 Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
%o A326199 v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
%o A326199 A326199(n) = v326199[n];
%Y A326199 Cf. A003557, A046523, A048250, A294877, A295300, A305801.
%Y A326199 Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.
%K A326199 nonn
%O A326199 1,2
%A A326199 _Antti Karttunen_, Jul 13 2019