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A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.

%I A374201 #10 Jul 02 2024 08:21:35
%S A374201 1,1,2,2,3,2,3,2,4,4,5,2,4,2,6,7,4,2,6,2,8,9,10,2,4,7,8,6,11,2,8,2,11,
%T A374201 8,12,7,4,2,12,13,7,2,11,2,8,8,13,2,7,10,7,8,14,2,15,7,13,12,10,2,8,2,
%U A374201 12,7,10,13,5,2,16,17,5,2,7,2,13,7,15,16,18,2,7,18,12,2,8,19,20,13,7,2,8,18,16,12,10,21,13,2,8,9,16
%N A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.
%C A374201 Restricted growth sequence transform of A278222(A048679(A328845(n))), or equally, of A304101(A328845(n)).
%C A374201 Related to the Zeckendorf-representation (A014417) of A328845(n).
%C A374201 For all i, j >= 0: a(i) = a(j) => A328847(i) = A328847(j).
%H A374201 Antti Karttunen, <a href="/A374201/b374201.txt">Table of n, a(n) for n = 0..75025</a>
%o A374201 (PARI)
%o A374201 up_to = 75025;
%o A374201 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 A374201 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A374201 A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
%o A374201 A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<<i; i++); n >>= 1); (s); };
%o A374201 A048679(n) = if(!n,n,A106151(2*A003714(n)));
%o A374201 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
%o A374201 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
%o A374201 A278222(n) = A046523(A005940(1+n));
%o A374201 A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
%o A374201 v374201 = rgs_transform(vector(1+up_to, n, A278222(A048679(A328845(n-1)))));
%o A374201 A374201(n) = v374201[1+n];
%Y A374201 Cf. A000045, A014417, A048679, A278222, A304101, A328845, A328847.
%Y A374201 Cf. also A304105, A373982.
%K A374201 nonn
%O A374201 0,3
%A A374201 _Antti Karttunen_, Jul 02 2024