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

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset

1,1

Comments

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.

The union of A211337 and A211338 is the complementary sequence to this one. - Douglas Latimer, Apr 12 2012

Numbers whose cubefree part (A050985) is not squarefree (A005117). - Amiram Eldar, Mar 09 2021

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.

L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Formula

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

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

A001248 UNION A030515 UNION A030627 UNION A030630 UNION A030633 UNION A030636 UNION ... - R. J. Mathar, May 05 2023

Example

a(7) = 28 is a term because the number of divisors of 28, d(28) = 6, is divisible by 3.

Maple

with(numtheory): for n from 1 to 1000 do if tau(n) mod 3 = 0 then printf(`%d, `, n) fi: od:

Mathematica

Select[Range[230], Divisible[DivisorSigma[0, #], 3] &] (* Amiram Eldar, Jul 26 2020 *)

Prog

(PARI) is(n)=vecmax(factor(n)[, 2]%3)==2 \\ Charles R Greathouse IV, Apr 10 2012
(PARI) is(n)=numdiv(n)%3==0 \\ Charles R Greathouse IV, Sep 18 2015

Crossrefs

Cf. A000005, A005117, A038109, A050985, A211337, A211338, A253905.

Characteristic function: A353470.

Sequence in context: A344275 A312862 A177880 * A369209 A081619 A377562

Adjacent sequences: A059266 A059267 A059268 * A059270 A059271 A059272

Keyword

nonn,easy

Author

Avi Peretz (njk(AT)netvision.net.il), Jan 24 2001

Extensions

More terms from James A. Sellers, Jan 24 2001

Status

approved