A133950 a(n) = the number of "isolated divisors" of n(n+1)/2. A positive divisor k of n is isolated if neither k-1 nor k+1 divides n.

1, 2, 1, 2, 4, 4, 4, 5, 6, 4, 5, 5, 4, 8, 10, 6, 6, 6, 6, 8, 8, 4, 8, 12, 6, 8, 11, 6, 8, 8, 8, 14, 8, 8, 14, 9, 4, 8, 16, 8, 8, 8, 6, 16, 12, 4, 12, 17, 9, 12, 13, 6, 8, 16, 18, 18, 8, 4, 11, 11, 4, 12, 28, 20, 16, 8, 6, 13, 16, 8, 14, 14, 4, 12, 19, 14, 16, 8, 12, 31, 10, 4, 11, 22, 8, 8, 18

Offset: 1

Author

Leroy Quet, Sep 30 2007

Keywords

• nonn

Examples

			a(8)=5 because 36 (=8*9/2) has 5 isolated divisors: 6,9,12,18,36.
		

Links

• Ray Chandler, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A133948, A133949, A063440.

Programs

Maple

with(numtheory): b:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1,div)=false and member(div[i]+1,div)=false then ISO:= `union`(ISO,{div[i]}) end if end do end proc: seq(nops(b((1/2)*j*(j+1))),j=1..80); # Emeric Deutsch, Oct 15 2007

Formula

a(n) = A063440(n) - A133949(n) = A132881(A000217(n)).

Extensions

More terms from Emeric Deutsch, Oct 15 2007

Extended by Ray Chandler, Jun 23 2008