A293899 - OEIS

%I #20 Nov 25 2023 07:53:11

%S 0,1,1,0,1,0,1,1,1,-1,1,1,1,1,0,0,1,0,1,1,2,-1,1,0,0,1,1,1,1,0,1,1,0,

%T -1,1,1,1,1,2,-1,1,0,1,1,0,-1,1,1,2,1,0,1,1,0,-1,1,2,-1,1,0,1,1,2,0,1,

%U 0,1,1,0,-1,1,0,1,1,1,1,1,0,1,1,1,-1,1,2,-1,1,0,-1,1,0,3,1,2,-1,1,0,1,1,0,0,1,0,1,1,0

%N Number of proper divisors of the form 3k+1 minus number of proper divisors of the form 3k+2.

%H Antti Karttunen, <a href="/A293899/b293899.txt">Table of n, a(n) for n = 1..20000</a>

%F When n = 3k, a(n) = A002324(n), when n = 3k+1, a(n) = A002324(n) - 1, when n = 3k+2, a(n) = A002324(n) + 1.

%F a(n) = A002324(n) - A010872(n) (mod 3).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - _Amiram Eldar_, Nov 25 2023

%t Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &] - DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* _Michael De Vlieger_, Nov 08 2017 *)

%t Table[Total[Which[Mod[#,3]==1,1,Mod[#,3]==2,-1,True,0]&/@Most[ Divisors[ n]]],{n,110}] (* _Harvey P. Dale_, Nov 26 2021 *)

%o (PARI)

%o A293895(n) = sumdiv(n,d,(d<n)*(1==(d%3)));

%o A293896(n) = sumdiv(n,d,(d<n)*(2==(d%3)));

%o A293899(n) = (A293895(n) - A293896(n));

%Y Cf. A002324, A010872, A073010, A293895, A293896.

%K sign,easy

%O 1,21

%A _Antti Karttunen_, Nov 06 2017