A159459 Rectangular array read by antidiagonals: a(n,m) = number of divisors of m that don't divide n.

0, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 3, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 2, 1, 2, 1, 3, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 0, 0, 1, 1, 1, 2, 0, 2, 2, 2, 1, 4, 1, 0, 1, 0, 2, 0, 2, 1, 3, 1, 2, 1, 4, 1, 3

Offset: 1

Author

Leroy Quet, Apr 12 2009

Keywords

• nonn
• tabl

Comments

a(n,1) = 0, for all n. a(1,m) = d(m)-1, for all m.

From Luc Rousseau, Jul 27 2018: (Start)

a(.,m) is periodic with period m.

a(n,m) is the number of nonzero elements S(n) and S(n+m) have in common, where S(n) denotes the set of complex numbers k*(1-exp(i*2*Pi*n/k)), for k positive integer. See illustration, section links.

(End)

Examples

			From _Andrew Howroyd_, Jul 27 2018: (Start)
Array begins:
  0 1 1 2 1 3 1 3 2 ...
  0 0 1 1 1 2 1 2 2 ...
  0 1 0 2 1 2 1 3 1 ...
  0 0 1 0 1 2 1 1 2 ...
  0 1 1 2 0 3 1 3 2 ...
  0 0 0 1 1 0 1 2 1 ...
  0 1 1 2 1 3 0 3 2 ...
  0 0 1 0 1 2 1 0 2 ...
  0 1 0 2 1 2 1 3 0 ...
  ...
(End) [corrected by _Jason Yuen_, Feb 05 2025]
		

Links

• Luc Rousseau, illustration, a(n,m) viewed as an intersection of sets of points

Crossrefs

Cf. A077478.

Programs

Maple

A159459 := proc(n,m) numtheory[tau](m)-numtheory[tau](gcd(n,m)) ; end: for d from 2 to 20 do for m from 1 to d-1 do n := d-m ; printf("%d,",A159459(n,m)) ; od: od: # R. J. Mathar, Apr 16 2009

Mathematica

Table[DivisorSigma[0, #] - DivisorSigma[0, GCD[n, #]] &[m - n + 1], {m, 13}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Jul 30 2018 *)

PARI

\\ port of R.J. Mathar's Maple program
a(n,m)=numdiv(m)-numdiv(gcd(n,m))
for(d=2,20,for(m=1,d-1,n=d-m;print1(a(n,m),", "))) \\ Luc Rousseau, Jul 27 2018

Formula

a(n,m) = d(m) - d(gcd(n,m)), where d(m) = A000005(m).

Extensions

2 terms corrected by R. J. Mathar, Apr 16 2009