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

%I A159459 #30 Feb 05 2025 09:22:23
%S A159459 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,
%T A159459 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,
%U A159459 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
%N A159459 Rectangular array read by antidiagonals: a(n,m) = number of divisors of m that don't divide n.
%C A159459 a(n,1) = 0, for all n. a(1,m) = d(m)-1, for all m.
%C A159459 From _Luc Rousseau_, Jul 27 2018: (Start)
%C A159459 a(.,m) is periodic with period m.
%C A159459 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.
%C A159459 (End)
%H A159459 Luc Rousseau, <a href="/A159459/a159459.png">illustration, a(n,m) viewed as an intersection of sets of points</a>
%F A159459 a(n,m) = d(m) - d(gcd(n,m)), where d(m) = A000005(m).
%e A159459 From _Andrew Howroyd_, Jul 27 2018: (Start)
%e A159459 Array begins:
%e A159459   0 1 1 2 1 3 1 3 2 ...
%e A159459   0 0 1 1 1 2 1 2 2 ...
%e A159459   0 1 0 2 1 2 1 3 1 ...
%e A159459   0 0 1 0 1 2 1 1 2 ...
%e A159459   0 1 1 2 0 3 1 3 2 ...
%e A159459   0 0 0 1 1 0 1 2 1 ...
%e A159459   0 1 1 2 1 3 0 3 2 ...
%e A159459   0 0 1 0 1 2 1 0 2 ...
%e A159459   0 1 0 2 1 2 1 3 0 ...
%e A159459   ...
%e A159459 (End) [corrected by _Jason Yuen_, Feb 05 2025]
%p A159459 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
%t A159459 Table[DivisorSigma[0, #] - DivisorSigma[0, GCD[n, #]] &[m - n + 1], {m, 13}, {n, m, 1, -1}] // Flatten (* _Michael De Vlieger_, Jul 30 2018 *)
%o A159459 (PARI) \\ port of R.J. Mathar's Maple program
%o A159459 a(n,m)=numdiv(m)-numdiv(gcd(n,m))
%o A159459 for(d=2,20,for(m=1,d-1,n=d-m;print1(a(n,m),", "))) \\ _Luc Rousseau_, Jul 27 2018
%Y A159459 Cf. A077478.
%K A159459 nonn,tabl
%O A159459 1,10
%A A159459 _Leroy Quet_, Apr 12 2009
%E A159459 2 terms corrected by _R. J. Mathar_, Apr 16 2009