A086145 Triangle read by rows in which T(n,k) is the least positive integer s such that p divides k^s-1, where p=prime(n) and k ranges from 1 to p-1.

1, 1, 2, 1, 4, 4, 2, 1, 3, 6, 3, 6, 2, 1, 10, 5, 5, 5, 10, 10, 10, 5, 2, 1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2, 1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2, 1, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2, 1, 11, 11, 11, 22, 11, 22, 11, 11, 22, 22, 11, 11, 22

Offset: 1

Author

Benoit Cloitre, Sep 06 2003

Keywords

• nonn
• tabf

Comments

The length of row n is A006093(n).

From J. H. Conway, Sep 06 2003: (Start)

"Let's ask for the exact power of some prime p that divides a^K - 1. Then the assertion is that if k is the smallest positive number for which p itself divides a^k - 1 and a^k - 1 is exactly divisible by p^i, then a^K - 1 will be divisible by p precisely when K is a multiple of k and then the exact power of p that divides it will be p^(i+j), where p^j is the exact power of p that divides K/k.

"In other words, the first time you get a multiple of p you can "accidentally" get a higher power than the first, but from then on you can only get more p's by putting them into the exponent.

"Examples: the first time 3^K - 1 is divisible by 11 is at 3^5 - 1, which is divisible precisely by 11^2. So 3^K - 1 will be divisible by 11^(2+j) only when KI is divisible by 5 times 11^j.

"Similarly, 2^1092 - 1 happens to be divisible by just 1093^2, so 2^(1092.1093^j) - 1 will be divisible by just 1093^(2+j)."

(End)

This is the prime-indexed rows of A057593. - Franklin T. Adams-Watters, Jan 19 2006

T(n,k) is the multiplicative order of k (mod prime(n)). Note that each row has many numbers that are the same. These numbers are counted in A174842. [T. D. Noe, Apr 01 2010]

Examples

			Triangle T(n,k) begins (with offsets 1):
[1]
[1, 2]
[1, 4, 4, 2]
[1, 3, 6, 3, 6, 2]
[1, 10, 5, 5, 5, 10, 10, 10, 5, 2]
[1, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2]
[1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2]
		

Links

• T. D. Noe, Rows n=1..50, flattened

Crossrefs

Cf. A006093, A057593, A174842.

Programs

Mathematica

Flatten[Table[MultiplicativeOrder[ #,p] & /@ Range[p-1], {p, Prime[Range[10]]}]] (* T. D. Noe, Apr 01 2010 *)

PARI

tabf(nn) = {for (n=1, nn, p = prime(n); for (k=1, p-1, print1(znorder(Mod(k, p)), ", ");); print(););} \\ Michel Marcus, Feb 05 2015

Extensions

Name improved by T. D. Noe, Apr 01 2010

Prepended 1 for p=2 by T. D. Noe, Apr 01 2010