cp's OEIS Frontend

A055212 Number of composite divisors of n.

%I A055212 #60 Dec 07 2023 01:43:55
%S A055212 0,0,0,1,0,1,0,2,1,1,0,3,0,1,1,3,0,3,0,3,1,1,0,5,1,1,2,3,0,4,0,4,1,1,
%T A055212 1,6,0,1,1,5,0,4,0,3,3,1,0,7,1,3,1,3,0,5,1,5,1,1,0,8,0,1,3,5,1,4,0,3,
%U A055212 1,4,0,9,0,1,3,3,1,4,0,7,3,1,0,8,1,1,1,5,0,8,1,3,1,1,1,9,0,3,3,6,0,4,0,5,4
%N A055212 Number of composite divisors of n.
%C A055212 Trivially, there is only one run of three consecutive 0's. However, there are infinitely many runs of three consecutive 1's and they are at positions A056809(n), A086005(n), and A115393(n) for n >= 1. - _Timothy L. Tiffin_, Jun 21 2021
%H A055212 Reinhard Zumkeller, <a href="/A055212/b055212.txt">Table of n, a(n) for n = 1..10000</a>
%F A055212 a(n) = A033273(n) - 1.
%F A055212 a(n) = tau(n)-omega(n)-1, where tau=A000005 and omega=A001221. - _Reinhard Zumkeller_, Jun 13 2003
%F A055212 G.f.: -x/(1 - x) + Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - _Ilya Gutkovskiy_, Mar 21 2017
%F A055212 Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 2 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - _Amiram Eldar_, Dec 07 2023
%e A055212 a[20] = 3 because the composite divisors of 20 are 4, 10, 20.
%t A055212 Table[ Count[ PrimeQ[ Divisors[n] ], False] - 1, {n, 1, 105} ]
%t A055212 Table[Count[Divisors[n],_?CompositeQ],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 09 2018 *)
%t A055212 a[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; Array[a, 100] (* _Amiram Eldar_, Jun 18 2022 *)
%o A055212 (Haskell)
%o A055212 a055212 = subtract 1 . a033273  -- _Reinhard Zumkeller_, Sep 15 2015
%o A055212 (PARI) a(n) = numdiv(n) - omega(n) - 1; \\ _Michel Marcus_, Oct 17 2015
%Y A055212 Complement of A083399.
%Y A055212 Cf. A000005, A001221, A008578 (indices of 0's), A033273, A056809, A086005, A115393, A137944, A137945, A344713.
%Y A055212 Cf. A001620, A077761.
%K A055212 easy,nonn
%O A055212 1,8
%A A055212 _Leroy Quet_, Jun 23 2000