1,2

Comment from John H Conway, Sep 06 2003:

"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)."

This is the prime-indexed rows of A057593, with an initial 1 added and the final 1 removed. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 19 2006

Triangle begins:

[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]

Sequence in context: A046943 A107728 A128250 this_sequence A113421 A135366 A051289

Adjacent sequences: A086142 A086143 A086144 this_sequence A086146 A086147 A086148

nonn,tabf

Benoit Cloitre, Sep 06 2003