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, 193, 199, 201, 208, 211, 217, 219, 223, 225, 228, 229

Offset: 1

Author

John L. Drost, Nov 25 2019

Keywords

• nonn

Comments

A number k is in the sequence iff in its prime factorization, all primes p == 1 (mod 3) occur to such a power p^e that e != 2 (mod 3), 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. (And here we don't have 196, although it is in A034022). - First sentence corrected and additional notes added by Antti Karttunen, Jul 03 2024, see also Robert Israel's Nov 09 2016 comment in A087943.

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. [Note: the "if and only if" condition given in the beginning of Theorem 7.1 is for A003136, not for this sequence. - _Antti Karttunen_, Jul 04 2024]
• Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Crossrefs

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

Cf. A000203, A353815 (characteristic function).

Setwise difference A003136 \ A088535.

Subsequences: A002476, A068228, A351537, A374135.

Cf. also A088232.

Not the same as A034022.

Programs

Magma

[k:k in [1..200]| DivisorSigma(1,k) mod 3 ne 0]; // Marius A. Burtea, Jan 02 2020

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 *)

PARI

isok(k) = (sigma(k) % 3) != 0; \\ Michel Marcus, Nov 26 2019

PARI

isA329963 = A353815; \\ Antti Karttunen, Jul 03 2024

Extensions

More terms from Joshua Oliver, Nov 26 2019

Data section further extended up to a(71), to better differentiate from nearby sequences - Antti Karttunen, Jul 04 2024