A211338 - OEIS

%I #44 Jan 06 2024 09:21:29

%S 2,3,5,7,11,13,16,17,19,23,24,29,30,31,37,40,41,42,43,47,53,54,56,59,

%T 61,66,67,70,71,73,78,79,81,83,88,89,97,101,102,103,104,105,107,109,

%U 110,113,114,127,128,130,131,135,136,137,138,139,149,151,152,154

%N Numbers k for which the number of divisors, tau(k), is congruent to 2 modulo 3.

%C The product of any 2 terms a(i)*a(j) is not a member of the sequence.

%C Any term a(n) can be expressed as 1 term (required to be greater than 1) from A211485 times 1 nonzero term from A000578. - _Douglas Latimer_, Apr 20 2012

%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 37, 368, 3681, 36596, 365336, 3653499, 36537962, 365381169, 3653826361, ... . Conjecture: the asymptotic density of this sequence exists and equals 3*zeta(3)/Pi^2 = 0.3653814847007... (A346602), so, a(n) ~ k*n with k = Pi^2/(3*zeta(3)) = 2.73686555524... . This conjecture is true if this sequence and A211337 have the same density (see A059269). - _Amiram Eldar_, Jan 06 2024

%H Amiram Eldar, <a href="/A211338/b211338.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Douglas Latimer)

%F Conjecture: a(n) ~ k*n where k = 2/prod(1 - (p-1)/(p^(3*k))) = 2.7290077... where p ranges over the primes and k ranges over the positive integers. - _Charles R Greathouse IV_, Apr 13 2012

%e The divisors of 16 are: 1, 2, 4, 8, 16 (5 divisors). 5 is congruent to 2 modulo 3. Thus 16 is a member of this sequence.

%t Select[Range[154], Mod[DivisorSigma[0, #], 3] == 2 &] (* _T. D. Noe_, Apr 21 2012 *)

%o (PARI) {plnt=1 ; mxind=100 ;for(k=1, 10^6,

%o if(numdiv(k) % 3 == 2, print(k); plnt++; if(mxind+1 == plnt, break() )))}

%Y This is an extension of A000040 (the prime numbers, which each have 2 divisors).

%Y The union of A059269 and A211337 is the complementary sequence to this one.

%Y The definition of this sequence uses A000005 (the number of divisors of n).

%Y Cf. A000578, A074796, A211485, A346602.

%K nonn

%O 1,1

%A _Douglas Latimer_, Apr 07 2012