A329963 Numbers k such that sigma(k) is not divisible by 3.

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192

Offset

1,2

Comments

A number k is in the sequence iff in its prime factorization, all primes == 1 (mod 3) occur to odd powers, and all primes == 2 (mod 3) occur to even powers. (3 can occur to any power.) This sequence is similar but not identical to many others; in particular, 343 is in this sequence, but not in A034022.

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 15 ff.

Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Maple

select(t -> numtheory:-sigma(t) mod 3 <> 0, [$1..200]); # Robert Israel, Jan 01 2020

Mathematica

Select[Range[200], !Divisible[DivisorSigma[1, #], 3] &] (* Amiram Eldar, Nov 25 2019 *)

Prog

(PARI) isok(k) = (sigma(k) % 3) != 0; \\ Michel Marcus, Nov 26 2019
(Magma) [k:k in [1..200]| DivisorSigma(1, k) mod 3 ne 0]; // Marius A. Burtea, Jan 02 2020

Crossrefs

Complement of A087943. Positions of zeros in A354100, nonzeros in A074941.

Subsequence of A003136.

Cf. A002144, A068228, A351537 (subsequences), A353815 (characteristic function).

Cf. A000203, A034022.

Cf. also A088232.

Sequence in context: A035238 A003136 A326421 * A034022 A198772 A185256

Adjacent sequences: A329960 A329961 A329962 * A329964 A329965 A329966

Keyword

nonn

Author

John L. Drost, Nov 25 2019

Extensions

More terms from Joshua Oliver, Nov 26 2019

Status

approved