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

1,10

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)

Links

Table of n, a(n) for n=1..105.

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

Formula

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

Example

From Andrew Howroyd, Jul 27 2018: (Start)
Array begins
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
  1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...
  1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 ...
  2 1 2 0 2 1 2 0 2 1 2 0 2 1 2 0 2 1 ...
  1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 ...
  3 2 2 2 3 0 3 2 2 2 3 0 3 2 2 2 3 0 ...
  1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 ...
  3 2 3 1 3 2 3 0 3 2 3 1 3 2 3 0 3 2 ...
  2 2 1 2 2 1 2 2 0 2 2 1 2 2 1 2 2 0 ...
  ...
(End)

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 *)

Prog

(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

Crossrefs

Cf. A077478.

Sequence in context: A357352 A339376 A362425 * A304095 A276675 A236389

Adjacent sequences: A159456 A159457 A159458 * A159460 A159461 A159462

Keyword

nonn,tabl

Author

Leroy Quet, Apr 12 2009

Extensions

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

Status

approved