A326395 - OEIS

%I #17 Jan 14 2024 02:28:07

%S 0,1,1,1,1,3,0,2,2,2,1,4,0,2,3,2,1,5,0,3,2,2,1,6,1,2,3,2,1,6,0,3,3,2,

%T 2,7,0,2,2,4,1,6,0,3,5,2,1,7,0,3,3,2,1,7,2,4,2,2,1,9,0,2,4,3,2,6,0,3,

%U 3,4,1,10,0,2,4,2,2,6,0,5,4,2,1,8,2,2,3,4,1,10

%N Expansion of Sum_{k>=1} x^(2*k) * (1 + x^k) / (1 - x^(3*k)).

%C Number of divisors of n that are not of the form 3*k + 1.

%H Robert Israel, <a href="/A326395/b326395.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(n) - A001817(n).

%F Sum_{k=1..n} a(k) ~ 2*n*log(n)/3 + c*n, where c = (5*gamma-2)/3 - gamma(1,3) = (5*A001620-2)/3 - A256425 = -0.382447... . - _Amiram Eldar_, Jan 14 2024

%p N:= 100: # for a(1) .. a(N)

%p S:= series(add(x^(2*k)*(1+x^k)/(1-x^(3*k)),k=1..N/2),x,N+1):

%p seq(coeff(S,x,i),i=1..N); # _Robert Israel_, Aug 27 2020

%t nmax = 90; CoefficientList[Series[Sum[x^(2 k) (1 + x^k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, 1 &, !MemberQ[{1}, Mod[#, 3]] &], {n, 1, 90}]

%o (PARI) a(n) = {numdiv(n) - sumdiv(n, d, d%3==1)} \\ _Andrew Howroyd_, Sep 11 2019

%Y Cf. A000005, A001817, A001822, A004611 (positions of 0's), A007494, A035191, A082050, A326394.

%Y Cf. A001620, A256425.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, Sep 11 2019