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Number of numbers k <= n such that tau(k) == 0 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.
3

%I #22 Aug 29 2020 06:00:08

%S 0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,4,4,5,5,5,5,5,6,6,6,7,7,7,7,8,8,8,

%T 8,9,9,9,9,9,9,9,9,10,11,11,11,11,12,13,13,14,14,14,14,14,14,14,14,15,

%U 15,15,16,16,16,16,16,17,17,17,17,18,18,18,19,20,20,20,20,20,20,20,20,21

%N Number of numbers k <= n such that tau(k) == 0 (mod 3) where tau(k) = A000005(k) is the number of divisors of k.

%H Amiram Eldar, <a href="/A074795/b074795.txt">Table of n, a(n) for n = 1..10000</a>

%H L. G. Sathe, <a href="https://www.jstor.org/stable/2371953">On a congruence property of the divisor function</a>, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

%F a(n) is asymptotic to c*n with c = 0.26....

%F The constant is c = 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305 ... (Sathe, 1945). - _Amiram Eldar_, Aug 29 2020

%t Accumulate[Boole[Divisible[DivisorSigma[0,Range[90]],3]]] (* _Harvey P. Dale_, Jan 11 2015 *)

%o (PARI) a(n)=sum(k=1,n,if(numdiv(k)%3,0,1))

%Y Cf. A000005, A059269, A074794, A074796.

%K nonn

%O 1,9

%A _Benoit Cloitre_, Sep 07 2002