A191257 - OEIS

A191257

a(n) = A067368(n)/2.

1, 3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 37, 39, 40, 41, 43, 45, 47, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 81, 83, 85, 87, 88, 89, 91, 93, 95, 97, 99, 101, 103, 104, 105, 107, 109, 111, 113, 115, 117, 119, 120, 121, 123, 125, 127, 129, 131, 133, 135, 136, 137, 139, 141, 143

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OFFSET

1,2

COMMENTS

From Jianing Song, Sep 21 2018: (Start)

Numbers n such that A191255(n) = 0 or 3. Previous definition was numbers n such that A191255(2*n) = 1, that is, numbers of the form 2^(3t)*s where s is an odd number.

{+-a(n)} gives all nonzero cubes modulo all powers of 2, that is, nonzero cubes over the 2-adic integers. So this sequence is closed under multiplication. (End)

The old entry had the conjecture that a(n) = A067368(n)/2. Jianing Song, Sep 21 2018 showed that this is true, and gave us the simpler definition that we have now used. The conjecture is correct because {a(n)} lists the numbers of the form 2^(3t)*s, and {A067368(n)} lists the numbers of the form 2^(3t+1)*s, where s is an odd number. Note also that a(n) = A213258(n)/4.

The asymptotic density of this sequence is 4/7. - Amiram Eldar, May 31 2024

LINKS

Table of n, a(n) for n=1..82.

Recto Rex M. Calingasan and Alexander Vincent B. Policarpio, On the zeros of the OEIS A191257 zeta function, AIP Conference Proceedings 1905, 030011 (2017).

MATHEMATICA

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3},

3 -> {0, 1}}] &, {0}, 9] (* A191255 *)

Flatten[Position[t, 0]] (* A005408, the odds *)

a = Flatten[Position[t, 1]] (* A067368 *)

b = Flatten[Position[t, 2]] (* A213258 *)

a/2 (* A191257 *)

b/4 (* a/2 *)

PROG

(PARI) isok(n) = valuation(2*n, 2)%3==1; \\ Altug Alkan, Sep 21 2018

CROSSREFS

Cf. A067368, A191255, A213258.

Perfect powers over the 2-adic integers:

Squares: positive: A234000; negative: A004215 (negated);

Cubes: this sequence;

Fourth powers: positive: A319281; negative: A319282 (negated).

Sequence in context: A153309 A047486 A229838 * A120212 A093670 A185011

Adjacent sequences: A191254 A191255 A191256 * A191258 A191259 A191260

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 28 2011

EXTENSIONS

Name corrected by Altug Alkan, Apr 03 2018

New name from Jianing Song, Sep 21 2018

STATUS

approved