A322418 Least k > 0 such that A014221(k) == A014221(k+1) mod n.

1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 4, 5, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 5, 6, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 5, 3, 4, 4, 3, 4, 3, 4, 5, 4, 5, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 5, 4, 5, 3, 4, 4, 4, 4, 5, 3

Offset

1,3

Comments

For any fixed integer n > 0, the sequence 2 mod n, 2^2 mod n, 2^2^2 mod n, that is, the sequence {A014221(i) mod n} for i >= 1 is eventually constant. a(n) is the least index k such that A014221(k) mod n equals this constant.

A038081(k+1) is the largest n such that a(n) = k.

Links

Table of n, a(n) for n=1..90.

USAMO, Problem 3, 1991.

Formula

a(n) <= A003434(n).

a(n) <= a(A000010(n)) + 1.

If A014221(k) == b(k) mod eulerphi(n), 0 < b(k) <= eulerphi(n), then a(n) is the least m > 0 such that 2^b(m-1) == 2^b(m) mod n.

Example

2, 4, 16, ... mod 6 equal 2, 4, 4, ..., so A014221(k) mod 6 = 4 for all k >= 2, hence a(6) = 2.

Prog

(PARI) a(n) = {c=0; k=1; x=1; d=n; while(k==1, z=x; y=1; b=1; while(z>0, while(y<z, d=eulerphi(d); y++); b=2^b-floor((2^b-1)/d)*d; z=z-1; y=1; d=n); if(c==b, k=0); c=b; x++); x-2; }

Crossrefs

Cf. A014221, A038081, A000010, A003434.

Sequence in context: A340862 A202472 A235613 * A019569 A003434 A330808

Adjacent sequences: A322415 A322416 A322417 * A322419 A322420 A322421

Keyword

nonn

Author

Jinyuan Wang, Dec 07 2018

Status

approved