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A323703 Number of values of (X^3 + X) mod prime(n). 1
1, 3, 3, 5, 7, 9, 11, 13, 15, 19, 21, 25, 27, 29, 31, 35, 39, 41, 45, 47, 49, 53, 55, 59, 65, 67, 69, 71, 73, 75, 85, 87, 91, 93, 99, 101, 105, 109, 111, 115, 119, 121, 127, 129, 131, 133, 141, 149, 151, 153, 155, 159, 161, 167, 171, 175, 179, 181, 185, 187, 189, 195 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the number of values of any other polynomial of degree 3, except X^3.

a(n) appears to approach (2/3)*prime(n) as n increases.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

For n >= 3, (3*a(n) - sin(a(n)*Pi/2))/2 = prime(n). - Murillo C. S. Fonseca, Nov 18 2019

For p = prime(n), floor((2*p + cos(p*Pi/log(p)))/3) <= a(n) <= ceiling((2*p - cos(p*Pi/log(p)))/3). - Pamela C. Lima, Feb 29 2020

a(n) = prime(n) - 2*floor(prime(n)/6 + 1/2), for n >= 3. - Ridouane Oudra, Jun 13 2020

EXAMPLE

a(1) = 1 since the only value X^3 + X takes mod 2 is 0.

MATHEMATICA

Array[Length@ Union@ Mod[Array[#^3 + # &, #], #] &@ Prime@ # &, 62] (* Michael De Vlieger, Jan 27 2019 *)

PROG

(PARI) a(n) = #Set(vector(prime(n), k, Mod(k^3+k, prime(n)))); \\ Michel Marcus, Jan 25 2019

CROSSREFS

Cf. A323704 (the number of values of X^3), A130291 (the number of values of X^2, which is also the number of values of any other polynomial of degree 2).

Sequence in context: A247130 A271974 A050824 * A333147 A323434 A323431

Adjacent sequences:  A323700 A323701 A323702 * A323704 A323705 A323706

KEYWORD

nonn

AUTHOR

Florian Severin, Jan 24 2019

STATUS

approved

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Last modified July 14 16:21 EDT 2020. Contains 335729 sequences. (Running on oeis4.)