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 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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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] LINKS T. D. Noe, Rows n=1..50, flattened EXAMPLE 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] MATHEMATICA Flatten[Table[MultiplicativeOrder[ #, p] & /@ Range[p-1], {p, Prime[Range[10]]}]] (* T. D. Noe, Apr 01 2010 *) PROG (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 CROSSREFS Cf. A006093, A057593, A174842. Sequence in context: A046943 A107728 A128250 * A261070 A249140 A113421 Adjacent sequences:  A086142 A086143 A086144 * A086146 A086147 A086148 KEYWORD nonn,tabf AUTHOR Benoit Cloitre, Sep 06 2003 EXTENSIONS Name improved by T. D. Noe, Apr 01 2010 Prepended 1 for p=2 by T. D. Noe, Apr 01 2010 STATUS approved

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Last modified May 19 08:25 EDT 2019. Contains 323389 sequences. (Running on oeis4.)