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.

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, 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, 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

Offset: 0

Author

Antti Karttunen, Jul 02 2024

Keywords

• nonn

Comments

Restricted growth sequence transform of A278222(A048679(A328845(n))), or equally, of A304101(A328845(n)).

Related to the Zeckendorf-representation (A014417) of A328845(n).

For all i, j >= 0: a(i) = a(j) => A328847(i) = A328847(j).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..75025

Crossrefs

Cf. A000045, A014417, A048679, A278222, A304101, A328845, A328847.

Cf. also A304105, A373982.

Programs

PARI

up_to = 75025;
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; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); };
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A278222(n) = A046523(A005940(1+n));
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]));
v374201 = rgs_transform(vector(1+up_to, n, A278222(A048679(A328845(n-1)))));
A374201(n) = v374201[1+n];