OFFSET
1,19
COMMENTS
T(n,1) = 0 since 1 | n^0.
T(n,p) = 1 for prime divisors p of n since p | n^1.
T(n,d) = 1 for divisors d > 1 of n since d | n^1.
Row n for prime p have maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.
Values greater than 1 pertain only to composite k of composite n > 4, but not in all cases. T(n,k) = 1 for squarefree kernels k of composite n.
T(n,k) = -1 for numbers k > 1 coprime to n and for numbers that are products of at least one prime q coprime to n and one prime p | n.
T(n,k) is nonnegative for all numbers k for which n^k (mod k) = 0, i.e., all the prime divisors p of k also divide n.
The largest possible value s in row n of T = floor(log_2(n)), since the largest possible multiplicity of any number m <= n pertains to perfect powers of 2, as 2 is the smallest prime. This number s first appears at T(2^s + 2, 2^s) for s > 1.
If T(n,k) is positive, 1/k terminates T(n,k) digits after the radix point in base n. If T(n,k) is negative, 1/k is recurrent in base n.
From Robert Israel, Dec 28 2016: (Start)
T(a*b,c*d) = max(T(a,c),T(b,d)) if GCD(a,b)=1, GCD(b,d)=1,T(a,c)>=0 and T(b,d)>=0.
T(n,a*b) = max(T(n,a),T(n,b)) if GCD(a,b)=1 and T(n,a)>=0 and T(n,b)>=0. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150)
EXAMPLE
The triangle T(n,k) begins (with -1 shown as "." for clarity):
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
1: 0
2: 0 1
3: 0 . 1
4: 0 1 . 1
5: 0 . . . 1
6: 0 1 1 2 . 1
7: 0 . . . . . 1
8: 0 1 . 1 . . . 1
9: 0 . 1 . . . . . 1
10: 0 1 . 2 1 . . 3 . 1
11: 0 . . . . . . . . . 1
12: 0 1 1 1 . 1 . 2 2 . . 1
13: 0 . . . . . . . . . . . 1
14: 0 1 . 2 . . 1 3 . . . . . 1
15: 0 . 1 . 1 . . . 2 . . . . . 1
...
MAPLE
f:= proc(n, k) local Fk, Fn, i;
if k = 1 then return 0 fi;
Fk:= ifactors(k)[2];
Fn:= map(t -> padic:-ordp(n, t[1]), Fk);
if min(Fn) = 0 then -1 else max(seq(ceil(Fk[i, 2]/Fn[i]), i=1..nops(Fk))) fi
end proc:
seq(seq(f(n, k), k=1..n), n=1..20); # Robert Israel, Dec 28 2016
MATHEMATICA
Table[Boole[k == 1] + (Boole[#[[-1, 1]] == 1] (-1 + Length@ #) /. 0 -> -1) &@ NestWhileList[Function[s, {#1/s, s}]@ GCD[#1, #2] & @@ # &, {k, n}, And[First@ # != 1, ! CoprimeQ @@ #] &], {n, 16}, {k, n}] // Flatten (* or *)
Table[SelectFirst[Range[0, Floor@ Log2@ n], PowerMod[n, #, k] == 0 &] /. k_ /; MissingQ@ k -> -1, {n, 12}, {k, n}] // TableForm (* Version 10.2 *)
CROSSREFS
KEYWORD
AUTHOR
Michael De Vlieger, Dec 26 2016
STATUS
approved