A123275 Square array A(n,m) = largest divisor of m which is coprime to n, read by upwards antidiagonals.

1, 1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 1, 1, 5, 1, 2, 3, 4, 5, 6, 1, 1, 3, 1, 5, 3, 7, 1, 2, 1, 4, 5, 2, 7, 8, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 3, 1, 5, 3, 1, 1, 9, 5, 11, 3, 13, 1, 2, 1, 4, 1, 2, 7, 8, 1, 2

Offset: 1

Author

Leroy Quet, Oct 10 2006

Keywords

• nonn
• tabl

Comments

Read by upwards antidiagonals as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), etc.

Seen as a triangle, the rows appear to be the reversed rows of the regular triangle defined by t(n,k) = denominator(n*k/(n-k)) for n>=2 and 1<=kMichel Marcus, Mar 24 2022

Examples

			The top left 18 x 18 corner of the array:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11,  4, 13, 14,  5, 16, 17,  2
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 3, 4, 1, 6, 7, 8, 9,  2, 11, 12, 13, 14,  3, 16, 17, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
  1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13,  2, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11,  4, 13, 14,  5, 16, 17,  2
  1, 1, 3, 1, 1, 3, 7, 1, 9,  1, 11,  3, 13,  7,  3,  1, 17,  9
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  1, 12, 13, 14, 15, 16, 17, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,  1, 14, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 1, 1, 9,  5, 11,  3, 13,  1, 15,  1, 17,  9
  1, 2, 1, 4, 1, 2, 7, 8, 1,  2, 11,  4, 13, 14,  1, 16, 17,  2
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,  1, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
...
A(12,1) = 12 because d=12 is the largest divisor of 12 for which gcd(d,1) = 1.
A(12,2) = 3 because d=3 is the largest divisor of 12 for which gcd(d,2) = 1.
A(12,3) = 4 because d=4 is the largest divisor of 12 for which gcd(d,3) = 1.
A(12,4) = 3 because d=3 is the largest divisor of 12 for which gcd(d,4) = 1.
A(12,6) = 1 because d=1 is the largest divisor of 12 for which gcd(d,6) = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Crossrefs

Cf. A003989, A020639, A243103.

Programs

Mathematica

t[n_, m_] := Last[Select[Divisors[m], GCD[ #, n] == 1 &]];Flatten[Table[t[i + 1 - j, j], {i, 15}, {j, i}]] (* Ray Chandler, Oct 17 2006 *)

Python

# Produces the triangle when the array is read by antidiagonals (upwards)
from sympy.ntheory import divisors
from math import gcd
def T(n,m):
    return [i for i in divisors(m) if gcd(i,n)==1][-1]
for i in range(1, 16):
    print([T(i+1-j, j) for j in range(1, i+1)]) # Indranil Ghosh, Mar 22 2017

Scheme

;; A naive implementation of A020639 given under that entry. The result of (A123275bi b a) is a product of all those prime factors of a (possibly occurring multiple times) that are not prime factors of b:
(define (A123275 n) (A123275bi (A004736 n) (A002260 n)))
(define (A123275bi b a) (let loop ((a a) (m 1)) (let ((s (A020639 a))) (cond ((= 1 a) m) ((zero? (modulo b s)) (loop (/ a s) m)) (else (loop (/ a s) (* s m)))))))
;; Antti Karttunen, Mar 22 2017

Extensions

Extended by Ray Chandler, Oct 17 2006

Name and Example section edited by Antti Karttunen, Mar 22 2017