A280269 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.

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, 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, 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

Offset

1,14

Comments

This table eliminates the negative values in row n of A279907.

Let k = A162306(n,m), i.e., the value in column m of row n.

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 two terms, {0,1}, the maximum value 1, since all k < p are coprime to p, and k | p^1 only when k = p.

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.

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.

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.

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.

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.

Originally from Robert Israel at A279907: (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..10202 (rows 1 <= n <= 660)

Example

Triangle T(n,m) begins:  Triangle A162306(n,k):
1:  0                    1
2:  0  1                 1  2
3:  0  1                 1  3
4:  0  1  1              1  2  4
5:  0  1                 1  5
6:  0  1  1  2  1        1  2  3  4  6
7:  0  1                 1  7
8:  0  1  1  1           1  2  4  8
9:  0  1  1              1  3  9
10: 0  1  2  1  3  1     1  2  4  5  8  10
...

Mathematica

Table[SelectFirst[Range[0, #], PowerMod[n, #, k] == 0 &] /. m_ /; MissingQ@ m -> Nothing &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* Version 10.2, or *)
DeleteCases[#, -1] & /@ Table[If[# == {}, -1, First@ #] &@ Select[Range[0, #], PowerMod[n, #, k] == 0 &] &@ Floor@ Log2@ n, {n, 24}, {k, n}] // Flatten (* or *)
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

Crossrefs

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.

Sequence in context: A258141 A037889 A339872 * A321894 A355824 A355826

Adjacent sequences: A280266 A280267 A280268 * A280270 A280271 A280272

Keyword

nonn,tabf,easy

Author

Michael De Vlieger, Dec 30 2016

Status

approved