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

%I A329963 #52 Jul 07 2024 13:41:03
%S A329963 1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,52,57,61,63,64,
%T A329963 67,73,75,76,79,81,84,91,93,97,100,103,108,109,111,112,117,121,124,
%U A329963 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
%N A329963 Numbers k such that sigma(k) is not divisible by 3.
%C A329963 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.
%C A329963 The asymptotic density of this sequence is 0 (Dressler, 1975). - _Amiram Eldar_, Jul 23 2020
%H A329963 Robert Israel, <a href="/A329963/b329963.txt">Table of n, a(n) for n = 1..10000</a>
%H A329963 Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, <a href="https://arxiv.org/abs/2007.03088">Arithmetic properties of the sum of divisors</a>, 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]
%H A329963 Robert E. Dressler, <a href="http://www.numdam.org/item/?id=CM_1975__31_2_115_0">A property of the phi and sigma_j functions</a>, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.
%p A329963 select(t -> numtheory:-sigma(t) mod 3 <> 0, [$1..200]); # _Robert Israel_, Jan 01 2020
%t A329963 Select[Range[200], !Divisible[DivisorSigma[1, #], 3] &] (* _Amiram Eldar_, Nov 25 2019 *)
%o A329963 (PARI) isok(k) = (sigma(k) % 3) != 0; \\ _Michel Marcus_, Nov 26 2019
%o A329963 (PARI) isA329963 = A353815; \\ _Antti Karttunen_, Jul 03 2024
%o A329963 (Magma) [k:k in [1..200]| DivisorSigma(1,k) mod 3 ne 0]; // _Marius A. Burtea_, Jan 02 2020
%Y A329963 Complement of A087943. Positions of zeros in A354100, nonzeros in A074941.
%Y A329963 Cf. A000203, A353815 (characteristic function).
%Y A329963 Setwise difference A003136 \ A088535.
%Y A329963 Subsequences: A002476, A068228, A351537, A374135.
%Y A329963 Cf. also A088232.
%Y A329963 Not the same as A034022.
%K A329963 nonn
%O A329963 1,2
%A A329963 _John L. Drost_, Nov 25 2019
%E A329963 More terms from _Joshua Oliver_, Nov 26 2019
%E A329963 Data section further extended up to a(71), to better differentiate from nearby sequences - _Antti Karttunen_, Jul 04 2024