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a(n) is the least k such that all possible modular classes a Fibonacci number can take mod n is seen in the Fibonacci numbers Fibonacci(1)..Fibonacci(k).
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%I #17 Dec 27 2024 00:57:40

%S 1,3,4,6,9,12,12,10,16,21,10,22,18,36,20,22,24,18,18,52,14,30,36,22,

%T 49,60,52,44,14,60,30,46,38,24,76,22,54,18,46,58,30,36,64,30,92,36,24,

%U 22,80,147,66,74,76,52,18,44,70,42,58,118,42,30,44,94,102,114,96

%N a(n) is the least k such that all possible modular classes a Fibonacci number can take mod n is seen in the Fibonacci numbers Fibonacci(1)..Fibonacci(k).

%C In verifying if k is in A367420 we only need to look from 1 to a(n) to see if there is a Fibonacci number f that has a remainder of k when dividing by 2*k.

%e The remainders of Fibonacci numbers mod 4 (starting at Fibonacci(1) = 1) are 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3. The distinct values are {0, 1, 2, 3}. The least k such that the remainders of Fibonacci numbers mod 4 contain all these values is 6 as the first 6 remainders are 1, 1, 2, 3, 1, 0.

%o (PARI)

%o a(n) = {if(n == 1, return(1));

%o my(rems = vector(n^2), v = [1,1]);

%o rems[1] = 1;

%o for(i = 2, n^2,

%o rems[i] = v[2];

%o v = [v[2], v[1]+v[2]]%n;

%o if(v == [1,1],

%o break

%o )

%o );

%o s = Set(rems);

%o for(i = 1, #rems,

%o s = setminus(s, Set(rems[i]));

%o if(#s == 0,

%o return(i)

%o )

%o )

%o }

%Y Cf. A000045, A001175, A001177, A189768, A367420.

%K nonn

%O 1,2

%A _David A. Corneth_, Nov 19 2023