A308972 Least k > 0 such that A114561(k) == A114561(k+1) mod n.

1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 3, 4, 2, 3, 2, 2, 1, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 3, 2, 4, 5, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 3, 4, 2, 4, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 4, 1, 2, 2, 2

Offset: 1

Author

Jinyuan Wang, Aug 30 2019

Keywords

• nonn

Comments

A114561(k+1) - A114561(k) is the largest n such that a(n) = k.

Examples

			4, 4^4, 4^4^4, ... mod 8 equal 4, 0, 0, ..., so A114561(k) mod 8 = 0 for all k >= 2, hence a(8) = 2.
		

Crossrefs

Cf. A114561, A308964, A322418.

Programs

PARI

a(n) = {c=0; k=1; x=0; d=n; while(k==1, z=x++; y=0; b=1; while(z>0, while(y++
				

Formula

a(n) <= A003434(n).

a(n) <= a(A000010(n)) + 1. Proof: a(n) <= a(eulerphi(n)) + 1. Proof: If A114561(i) == b(i) mod eulerphi(n), 0 < b(i) <= eulerphi(n), then a(n) is the least k > 0 such that 2^b(k-1) == 2^b(k) mod n. Since A114561(a(eulerphi(n))) == A114561(a(eulerphi(n)) + 1), k <= a(A000010(n)) + 1.