A129510 Number of distinct differences between pairs of distinct divisors of n.

0, 1, 1, 3, 1, 5, 1, 6, 3, 6, 1, 10, 1, 6, 5, 10, 1, 13, 1, 13, 6, 6, 1, 18, 3, 6, 6, 14, 1, 19, 1, 15, 6, 6, 6, 24, 1, 6, 6, 22, 1, 23, 1, 15, 12, 6, 1, 30, 3, 15, 6, 15, 1, 25, 6, 23, 6, 6, 1, 37, 1, 6, 13, 21, 6, 25, 1, 15, 6, 24, 1, 40, 1, 6, 13, 15, 6, 26, 1, 34, 10, 6, 1, 45, 6, 6, 6, 26

Offset: 1

Author

Reinhard Zumkeller, Apr 19 2007

Keywords

• nonn

Comments

a(n) = #{|x-y|: x <> y and n mod x = n mod y = 0};

a(n) = 1 iff n is prime;

a(n) <= A000217(A000005(a(n))-1) = A066446(n):

a(A129511(n))=A066446(A129511(n)), a(A129512(n)) < A066446(A129512(n)).

Examples

			n=44, set of divisors of 44 = {1,2,4,11,22,44}:
44-22=22, 44-11=33, 44-4=40, 44-2=42, 44-1=41,
22-11=11, 22-4=18, 22-2=20, 22-1=21,
11-4=7, 11-2=9, 11-1=10, 4-2=2, 4-1=3, 2-1=1,
a(44) = #{1,2,3,7,9,10,11,18,20,21,22,33,40,41,42} = 15;
n=45, set of divisors of 45 = {1,3,5,9,15,45}:
45-15=30, 45-9=36, 45-5=40, 45-3=42, 45-1=44,
15-9=6, 15-5=10, 15-3=12, 15-1=14,
9-5=4, 9-3=6, 9-1=8, 5-3=2, 5-1=4, 3-1=2,
a(45) = #{2,4,6,8,10,12,14,30,36,40,42,44} = 12.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor

Crossrefs

Cf. A000005.

Programs

Mathematica

a[n_]:=Length[Union[Flatten[Differences/@Subsets[Divisors[n],{2}]]]];Table[a[n],{n,88}] (* James C. McMahon, Jan 21 2025 *)

PARI

a(n)=my(d=divisors(n),v=List()); for(i=1,#d-1,for(j=i+1,#d, listput(v,d[j]-d[i]))); #Set(v) \\ Charles R Greathouse IV, Aug 26 2015