A355831 - OEIS

%I #7 Jul 20 2022 22:38:41

%S 1,2,2,3,2,4,2,5,6,4,2,7,2,4,4,8,2,9,2,7,10,4,2,11,12,10,13,7,2,14,2,

%T 15,4,4,16,17,2,4,4,18,2,19,2,20,21,10,2,22,6,23,10,24,2,25,4,18,4,4,

%U 2,26,2,4,27,28,16,14,2,7,4,29,2,30,2,4,31,32,16,19,2,33,34,4,2,35,4,10,4,36,2,37,4,38,4,39,16,40,2,41,21,42,2,43,2,44,45

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A354347(i) = A354347(j) for all i, j >= 1.

%C Restricted growth sequence transform of the ordered pair [A046523(n), A354347(n)].

%H Antti Karttunen, <a href="/A355831/b355831.txt">Table of n, a(n) for n = 1..65537</a>

%o (PARI)

%o up_to = 65537;

%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 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A345000(n) = gcd(A003415(n),A003415(A276086(n)));

%o v354347 = DirInverseCorrect(vector(up_to,n,A345000(n)));

%o A354347(n) = v354347[n];

%o Aux355831(n) = [A046523(n), A354347(n)];

%o v355831 = rgs_transform(vector(up_to,n,Aux355831(n)));

%o A355831(n) = v355831[n];

%Y Cf. A305800, A355830, A355832.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jul 20 2022