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%I #16 Jul 15 2024 03:21:13
%S 0,0,1,0,1,0,1,1,0,1,0,1,1,2,1,0,1,0,1,1,1,0,1,1,0,1,2,1,3,1,0,1,0,1,
%T 1,1,1,2,2,1,0,1,0,1,2,1,3,1,0,1,1,2,1,0,1,1,1,1,0,1,0,1,1,2,1,3,1,2,
%U 4,1,0,1,0,1,1,1,2,1,2,1,0,1,1,2,1,0,1,2,3,1,4,1,0,1,0,1,1,1,1,1
%N Irregular triangle T(n,m) read by rows: smallest power e of n that is divisible by m = term k in row n of A162306.
%C This table eliminates the negative values in row n of A279907.
%C Let k = A162306(n,m), i.e., the value in column m of row n.
%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 two terms, {0,1}, the maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.
%C Row n for prime power p^i have (i+1) terms, one zero and i ones, since all k that appear in corresponding row n of A162306 are divisors d of p^i.
%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 Numbers k > 1 coprime to n and numbers that are products of at least one prime q coprime to n and one prime p | n do not appear in A162306; these do not divide n^e evenly.
%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 1/k terminates T(n,k) digits after the radix point in base n for values of k that appear in row n of A162306.
%C Originally from _Robert Israel_ at A279907: (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.
%C (End)
%H Michael De Vlieger, <a href="/A280269/b280269.txt">Table of n, a(n) for n = 1..10202</a> (rows 1 <= n <= 660)
%e Triangle T(n,m) begins: Triangle A162306(n,k):
%e 1: 0 1
%e 2: 0 1 1 2
%e 3: 0 1 1 3
%e 4: 0 1 1 1 2 4
%e 5: 0 1 1 5
%e 6: 0 1 1 2 1 1 2 3 4 6
%e 7: 0 1 1 7
%e 8: 0 1 1 1 1 2 4 8
%e 9: 0 1 1 1 3 9
%e 10: 0 1 2 1 3 1 1 2 4 5 8 10
%e ...
%t Table[SelectFirst[Range[0, #], PowerMod[n, #, k] == 0 &] /. m_ /; MissingQ@ m -> Nothing &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* Version 10.2, or *)
%t DeleteCases[#, -1] & /@ Table[If[# == {}, -1, First@ #] &@ Select[Range[0, #], PowerMod[n, #, k] == 0 &] &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* or *)
%t DeleteCases[#, -1] & /@ 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, 24}, {k, n}] // Flatten
%Y Cf. A162306, A279907 (T(n,k) with values for all 1 <= k <= n), A280274 (maximum values in row n), A010846 (number of nonnegative k in row n), A051731 (k with e <= 1), A000005 (number of k in row n with e <= 1), A272618 (k with e > 1), A243822 (number of k in row n with e > 1), A007947.
%K nonn,tabf,easy
%O 1,14
%A _Michael De Vlieger_, Dec 30 2016
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