A351537 -id:A351537 - cp's OEIS frontend

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

John L. Drost, Nov 25 2019

nonn

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

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.

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.

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

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

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

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

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

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

A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

Original entry on oeis.org

26, 74, 122, 146, 194, 218, 234, 314, 362, 386, 458, 482, 554, 626, 650, 666, 674, 698, 746, 794, 818, 842, 866, 914, 1082, 1098, 1154, 1202, 1226, 1314, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1746, 1754, 1850, 1874, 1962, 1994, 2018, 2042, 2066, 2106, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2498

Offset: 1

Antti Karttunen, Feb 14 2022

nonn

All the terms of the form 4u+2 in A349745 (if they exist) are found in this sequence. As A351537 is the intersection of A191218 and A329963, and the latter has asymptotic density zero, so has this sequence also. It is conjectured that A351555(a(n)) is nonzero for all n, which would imply that the intersection with A349745 is empty. - Antti Karttunen, Feb 19 2022

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences related to sigma(n)

Cf. A191218, A329963, A349745, A351537.

Probably a subsequence of A351543. (See also A351550, A351555).

isA351538(n) = if(!(2==(n%4)),0, my(s=sigma(n)); (2 == (s%4)) && (1==valuation(s,3)));

a(n) = 2 * A351537(n).

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

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A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

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