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A323401 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

%I A323401 #8 Jan 15 2019 18:42:43
%S A323401 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 A323401 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 A323401 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 A323401 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.
%C A323401 Restricted growth sequence transform of function f, defined as f(n) = A323372(n) for all other numbers n, except f(p) = 0 for odd primes p.
%C A323401 For all i, j:
%C A323401   A323400(i) = A323400(j) => a(i) = a(j),
%C A323401   a(i) = a(j) => A322588(i) = A322588(j),
%C A323401   a(i) = a(j) => A323364(i) = A323364(j).
%H A323401 Antti Karttunen, <a href="/A323401/b323401.txt">Table of n, a(n) for n = 1..65537</a>
%o A323401 (PARI)
%o A323401 up_to = 65537;
%o A323401 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 A323401 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -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 A323401 A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
%o A323401 v323363 = DirInverse(vector(up_to,n,A001615(n)));
%o A323401 A323363(n) = v323363[n];
%o A323401 A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
%o A323401 Aux323401(n) = if((n>2)&&isprime(n), 0, [A003557(n), A323363(n)]);
%o A323401 v323401 = rgs_transform(vector(up_to, n, Aux323401(n)));
%o A323401 A323401(n) = v323401[n];
%Y A323401 Cf. A001615, A322588, A323363, A323364, A323400,
%Y A323401 Cf. also A323371, A323372.
%K A323401 nonn
%O A323401 1,2
%A A323401 _Antti Karttunen_, Jan 15 2019