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A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

%I A323370 #7 Jan 13 2019 19:29:13
%S A323370 1,2,3,4,3,5,3,6,7,8,3,9,3,10,11,12,3,13,3,14,15,16,3,17,18,19,20,21,
%T A323370 3,22,3,23,24,25,26,27,3,28,26,29,3,30,3,31,32,33,3,34,35,36,37,38,3,
%U A323370 39,40,41,42,43,3,44,3,45,46,47,48,49,3,50,51,52,3,53,3,54,55,56,57,52,3,58,59,60,3,61,62,63,64,65,3,66,67,68,57,69,67,70,3,71,72,73,3,74,3,75,76
%N A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.
%C A323370 Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers.
%C A323370 For all i, j:
%C A323370   A305801(i) = A305801(j) => a(i) = a(j),
%C A323370   a(i) = a(j) => A323367(i) = A323367(j),
%C A323370   a(i) = a(j) => A323371(i) = A323371(j).
%H A323370 Antti Karttunen, <a href="/A323370/b323370.txt">Table of n, a(n) for n = 1..65537</a>
%o A323370 (PARI)
%o A323370 up_to = 65537;
%o A323370 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 A323370 A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
%o A323370 A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
%o A323370 Aux323370(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A023900(n)]);
%o A323370 v323370 = rgs_transform(vector(up_to, n, Aux323370(n)));
%o A323370 A323370(n) = v323370[n];
%Y A323370 Cf. A000035,  A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323367, A323371.
%Y A323370 Differs from A323405 for the first time at n=78, where a(78) = 52, while A323405(78) = 58.
%K A323370 nonn
%O A323370 1,2
%A A323370 _Antti Karttunen_, Jan 13 2019