cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

John L. Drost, Nov 25 2019

Keywords

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

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

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

A070563 a(n) = 0 if 3 divides the Ramanujan number tau(n) (A000594(n)), otherwise 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
Offset: 1

Views

Author

N. J. A. Sloane, May 07 2002

Keywords

Comments

Multiplicative because A000594 is. Conjecture: a(3^k) = 0, if p == 1 mod 3, a(p^2k) = 0 and a(p^(2k+1)) = 1, if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. - Christian G. Bower, Jun 10 2005
From Antti Karttunen, Jul 03 2024: (Start)
The above conjecture is not correct. The first counterexample occurs at n = 2401 = 7^4. My improved conjecture is that this is actually a characteristic function of nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3, that is, having a following multiplicative formula: a(3^k) = 0, if p == 1 mod 3, a(p^e) = 1 if e != 2 (mod 3), otherwise 0, and if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. This conjecture has now been proved correct by Seiichi Manyama.
Bower's formula is now submitted as A374053.
(End)

Links

Crossrefs

Characteristic function of A374135, nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3.
Cf. A000203, A000594, A011655, A070564, A126825, A353815, A374053.

Programs

Formula

a(n) = A011655(n) * A353815(n), conjectured by Antti Karttunen, proved by Seiichi Manyama, Jul 03 2024
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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