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%I #32 Jun 27 2021 03:39:59
%S 0,0,1,0,-1,1,0,1,-1,1,0,-1,-1,-1,1,0,1,1,2,-1,1,0,-1,-1,-1,-1,-1,1,0,
%T 1,-1,1,-1,-1,-1,1,0,-1,1,-1,-1,-1,-1,-1,1,0,1,-1,2,1,-1,-1,3,-1,1,0,
%U -1,-1,-1,-1,-1,-1,-1,-1,-1,1,0,1,1,1,-1,1,-1,2,2,-1,-1,1,0,-1,-1,-1,-1,-1,-1
%N Triangle read by rows: T(n,k) is the smallest power of n that is divisible by k, or -1 if no such power exists.
%C T(n,1) = 0 since 1 | n^0.
%C T(n,p) = 1 for prime divisors p of n since p | n^1.
%C T(n,d) = 1 for divisors d > 1 of n since d | n^1.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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.
%C From _Robert Israel_, Dec 28 2016: (Start)
%C 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.
%C 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)
%H Michael De Vlieger, <a href="/A279907/b279907.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150)
%e The triangle T(n,k) begins (with -1 shown as "." for clarity):
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
%e 1: 0
%e 2: 0 1
%e 3: 0 . 1
%e 4: 0 1 . 1
%e 5: 0 . . . 1
%e 6: 0 1 1 2 . 1
%e 7: 0 . . . . . 1
%e 8: 0 1 . 1 . . . 1
%e 9: 0 . 1 . . . . . 1
%e 10: 0 1 . 2 1 . . 3 . 1
%e 11: 0 . . . . . . . . . 1
%e 12: 0 1 1 1 . 1 . 2 2 . . 1
%e 13: 0 . . . . . . . . . . . 1
%e 14: 0 1 . 2 . . 1 3 . . . . . 1
%e 15: 0 . 1 . 1 . . . 2 . . . . . 1
%e ...
%p f:= proc(n,k) local Fk,Fn,i;
%p if k = 1 then return 0 fi;
%p Fk:= ifactors(k)[2];
%p Fn:= map(t -> padic:-ordp(n,t[1]),Fk);
%p if min(Fn) = 0 then -1 else max(seq(ceil(Fk[i,2]/Fn[i]),i=1..nops(Fk))) fi
%p end proc:
%p seq(seq(f(n,k),k=1..n),n=1..20); # _Robert Israel_, Dec 28 2016
%t 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 *)
%t Table[SelectFirst[Range[0, Floor@ Log2@ n], PowerMod[n, #, k] == 0 &] /. k_ /; MissingQ@ k -> -1, {n, 12}, {k, n}] // TableForm (* Version 10.2 *)
%Y Cf.: A010846 (number of nonnegative k in row n), A162306 (k with nonnegative values in a(n)), A051731 (k with values 0 or 1), A000005 (number of k in row n with values 0 or 1), A272618 (k with values > 1), A243822 (number of k in row n with values > 1), A007947.
%K tabl,sign,easy
%O 1,19
%A _Michael De Vlieger_, Dec 26 2016
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