cp's OEIS Frontend

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

%I A129510 #9 Feb 16 2025 08:33:05
%S A129510 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,
%T A129510 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,
%U A129510 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
%N A129510 Number of distinct differences between pairs of distinct divisors of n.
%C A129510 a(n) = #{|x-y|: x <> y and n mod x = n mod y = 0};
%C A129510 a(n) = 1 iff n is prime;
%C A129510 a(n) <= A000217(A000005(a(n))-1) = A066446(n):
%C A129510 a(A129511(n))=A066446(A129511(n)), a(A129512(n)) < A066446(A129512(n)).
%H A129510 Charles R Greathouse IV, <a href="/A129510/b129510.txt">Table of n, a(n) for n = 1..10000</a>
%H A129510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%e A129510 n=44, set of divisors of 44 = {1,2,4,11,22,44}:
%e A129510 44-22=22, 44-11=33, 44-4=40, 44-2=42, 44-1=41,
%e A129510 22-11=11, 22-4=18, 22-2=20, 22-1=21,
%e A129510 11-4=7, 11-2=9, 11-1=10, 4-2=2, 4-1=3, 2-1=1,
%e A129510 a(44) = #{1,2,3,7,9,10,11,18,20,21,22,33,40,41,42} = 15;
%e A129510 n=45, set of divisors of 45 = {1,3,5,9,15,45}:
%e A129510 45-15=30, 45-9=36, 45-5=40, 45-3=42, 45-1=44,
%e A129510 15-9=6, 15-5=10, 15-3=12, 15-1=14,
%e A129510 9-5=4, 9-3=6, 9-1=8, 5-3=2, 5-1=4, 3-1=2,
%e A129510 a(45) = #{2,4,6,8,10,12,14,30,36,40,42,44} = 12.
%t A129510 a[n_]:=Length[Union[Flatten[Differences/@Subsets[Divisors[n],{2}]]]];Table[a[n],{n,88}] (* _James C. McMahon_, Jan 21 2025 *)
%o A129510 (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
%Y A129510 Cf. A000005.
%K A129510 nonn
%O A129510 1,4
%A A129510 _Reinhard Zumkeller_, Apr 19 2007