A133951 a(n) is the number of "non-isolated divisors" of n!. A positive divisor k of n is non-isolated if either k-1 or k+1 also divides n.

0, 2, 3, 4, 6, 11, 17, 19, 23, 27, 43, 43, 64, 74, 80, 82, 124, 124, 177, 185, 195, 214, 300, 300, 300, 328, 328, 334, 454, 454, 618, 618, 635, 677, 677, 677, 872, 936, 949, 949, 1224, 1228, 1579, 1587, 1587, 1672, 2124, 2124, 2126, 2126, 2148, 2154, 2707, 2707, 2709, 2709

Offset: 1

Author

Leroy Quet, Sep 30 2007

Keywords

• nonn

Examples

			a(6)=11 because 1,2,3,4,5,6,8,9,10,15,16 are the non-isolated divisors of 720.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..60

Crossrefs

Cf. A000142, A027423, A132747, A133952.

Programs

Maple

with(numtheory): A:=proc(n) local div, NID, i: div:=divisors(factorial(n)): NID:={}: for i to tau(factorial(n)) do if member(div[i]-1, div)=true or member(div[i]+1, div)=true then NID:= `union`(NID, {div[i]}) else end if end do: NID end proc: seq(nops(A(n)),n=1..30); # Emeric Deutsch, Oct 12 2007

Formula

a(n) = A027423(n) - A133952(n) = A132747(A000142(n)).

Extensions

Corrected and extended by Emeric Deutsch, Oct 12 2007

a(31)-a(35) from Ray Chandler, May 28 2008

a(36)-a(50) from Ray Chandler, Jun 20 2008

a(51)-a(56) from Lucas A. Brown, Oct 02 2024