cp's OEIS Frontend

A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

%I A059269 #42 Jul 02 2025 16:02:00
%S A059269 4,9,12,18,20,25,28,32,36,44,45,49,50,52,60,63,68,72,75,76,84,90,92,
%T A059269 96,98,99,100,108,116,117,121,124,126,132,140,144,147,148,150,153,156,
%U A059269 160,164,169,171,172,175,180,188,196,198,200,204,207,212,220,224,225,228
%N A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.
%C A059269 tau(n) is divisible by 3 iff at least one prime in the prime factorization of n has exponent of the form 3*m + 2. This sequence is an extension of the sequence A038109 in which the numbers has at least one prime with exponent 2 (the case of m = 0 here ) in their prime factorization.
%C A059269 The union of A211337 and A211338 is the complementary sequence to this one. - _Douglas Latimer_, Apr 12 2012
%C A059269 Numbers whose cubefree part (A050985) is not squarefree (A005117). - _Amiram Eldar_, Mar 09 2021
%H A059269 Charles R Greathouse IV, <a href="/A059269/b059269.txt">Table of n, a(n) for n = 1..10000</a>
%H A059269 Eckford Cohen, <a href="https://eudml.org/doc/140760">Arithmetical Notes, XIII. A Sequal to Note IV</a>, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
%H A059269 S. S. Pillai, <a href="https://doi.org/10.18311/jims/1942/17182">On a congruence property of the divisor function</a>, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
%H A059269 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 A059269 Conjecture: a(n) ~ k*n where k = 1/(1 - Product(1 - (p-1)/(p^(3*i)))) = 3.743455... where p ranges over the primes and i ranges over the positive integers. - _Charles R Greathouse IV_, Apr 13 2012
%F A059269 The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305... (Sathe, 1945). Therefore, the above conjecture, a(n) ~ k*n, is true, but k = 1/(1-6*zeta(3)/Pi^2) = 3.7141993349... - _Amiram Eldar_, Jul 26 2020
%F A059269 A001248 UNION A030515 UNION A030627 UNION A030630 UNION A030633 UNION A030636 UNION ... - _R. J. Mathar_, May 05 2023
%e A059269 a(7) = 28 is a term because the number of divisors of 28, d(28) = 6, is divisible by 3.
%p A059269 with(numtheory): for n from 1 to 1000 do if tau(n) mod 3 = 0 then printf(`%d,`,n) fi: od:
%t A059269 Select[Range[230], Divisible[DivisorSigma[0, #], 3] &] (* _Amiram Eldar_, Jul 26 2020 *)
%o A059269 (PARI) is(n)=vecmax(factor(n)[,2]%3)==2 \\ _Charles R Greathouse IV_, Apr 10 2012
%o A059269 (PARI) is(n)=numdiv(n)%3==0 \\ _Charles R Greathouse IV_, Sep 18 2015
%Y A059269 Cf. A000005, A005117, A038109, A050985, A211337, A211338, A253905.
%Y A059269 Characteristic function: A353470.
%K A059269 nonn,easy
%O A059269 1,1
%A A059269 Avi Peretz (njk(AT)netvision.net.il), Jan 24 2001
%E A059269 More terms from _James Sellers_, Jan 24 2001