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

1,5

Comments

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

Links

Table of n, a(n) for n=1..87.

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.

Example

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.

Prog

(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++<z, d=eulerphi(d)); b=4^b-floor((4^b-1)/d)*d; z=z-1; y=0; d=n); if(c==b, k=0); c=b); x-1; }

Crossrefs

Cf. A114561, A308964, A322418.

Sequence in context: A257540 A333516 A228202 * A137900 A025837 A111317

Adjacent sequences: A308969 A308970 A308971 * A308973 A308974 A308975

Keyword

nonn

Author

Jinyuan Wang, Aug 30 2019

Status

approved