%I #6 Sep 05 2019 20:49:15

%S 1,2,2,2,2,3,2,2,2,4,2,5,2,4,6,2,2,7,2,8,2,4,2,9,2,4,2,5,2,7,2,2,6,4,

%T 2,4,2,4,2,4,2,7,2,5,10,4,2,5,2,4,6,4,2,7,2,11,2,4,2,12,2,4,2,2,2,7,2,

%U 4,6,4,2,13,2,4,2,5,2,7,2,4,2,4,2,5,2,4,6,5,2,14,15,5,2,4,16,9,2,4,6,8,2,7,2,4,6

%N Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = gcd(n,usigma(n)) * (-1)^[gcd(n,usigma(n))==n], and usigma is the sum of unitary divisors of n (A034448).

%C Restricted growth sequence transform of function f, defined as f(n) = -A323166(n) = -n when n is one of unitary multiply-perfect numbers (A327158), otherwise f(n) = A323166(n) = gcd(n,A034448(n))

%C For all i, j:

%C A305800(i) = A305800(j) => a(i) = a(j) => A327164(i) = A327164(j).

%H Antti Karttunen, <a href="/A327163/b327163.txt">Table of n, a(n) for n = 1..87360</a>

%o (PARI)

%o up_to = 87360;

%o 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 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A323166(n) = gcd(n, A034448(n));

%o Aux327163(n) = { my(u=A323166(n)); u*((-1)^(u==n)); };

%o v327163 = rgs_transform(vector(up_to, n, Aux327163(n)));

%o A327163(n) = v327163[n];

%Y Cf. A034448, A305800, A323166, A327158, A327164.

%Y Cf. also A326193.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 28 2019