cp's OEIS Frontend

A340227 Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.

%I A340227 #9 Feb 08 2025 13:58:45
%S A340227 0,1,1,1,1,4,1,1,1,4,1,4,1,4,4,1,1,4,1,4,4,4,1,4,1,4,1,4,1,13,1,1,4,4,
%T A340227 4,4,1,4,4,4,1,13,1,4,4,4,1,4,1,4,4,4,1,4,4,4,4,4,1,13,1,4,4,1,4,13,1,
%U A340227 4,4,13,1,4,1,4,4,4,4,13,1,4,1,4,1,13,4,4,4,4,1,13
%N A340227 Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.
%C A340227 If n = p where p is prime, the only pair of divisors of n such that d1 < d2 is (1,p). Since the product 1*p = p is squarefree, this satisfies the constraints. Thus, a(p) = 1 for all p. - _Wesley Ivan Hurt_, May 21 2021
%F A340227 Sum_{d1|n, d2|n, d1<d2} mu(d1*d2)^2, where mu is the Möbius function (A008683).
%F A340227 a(n) = (A074816(n) - 1)/2. - _Ridouane Oudra_, Jan 27 2025
%e A340227 a(28) = 4; (1,2), (1,7), (1,14), (2,7)
%e A340227 a(29) = 1; (1,29)
%e A340227 a(30) = 13; (1,2), (1,3), (1,5), (1,6), (1,10), (1,15), (1,30), (2,3), (2,5), (2,15), (3,5), (3,10), (5,6)
%e A340227 a(31) = 1; (1,31)
%p A340227 with(numtheory): seq((3^nops(factorset(n))-1)/2, n=1..100); # _Ridouane Oudra_, Jan 27 2025
%t A340227 Table[Sum[Sum[MoebiusMu[i*k]^2 (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]
%Y A340227 Cf. A008683, A074816.
%K A340227 nonn
%O A340227 1,6
%A A340227 _Wesley Ivan Hurt_, Jan 01 2021