A332020 Positive integers m which are quadratic residues modulo prime(m).

1, 4, 5, 9, 12, 14, 16, 17, 19, 20, 22, 23, 25, 29, 30, 31, 34, 35, 36, 37, 38, 40, 42, 43, 46, 47, 49, 51, 53, 57, 59, 61, 63, 64, 66, 67, 70, 72, 73, 76, 77, 78, 80, 81, 82, 86, 87, 89, 91, 92, 94, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 111, 112, 113, 115, 121, 125, 127, 128, 132, 134, 136, 137, 138, 140

Offset

1,2

Comments

Clearly, the sequence contains all positive squares.

Conjecture: Let A(x) be the number of terms not exceeding x. Then A(x)/x has the limit 1/2 as x tends to the infinity.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 since 1 is a quadratic residue modulo prime(1) = 2.
a(2) = 4 since 4 is a quadratic residue modulo prime(4) = 7, but 2 is a quadratic nonresidue modulo prime(2) = 3, and 3 is a quadratic nonresidue modulo prime(3) = 5.

Mathematica

tab = {}; Do[If[JacobiSymbol[n, Prime[n]] == 1, tab = Append[tab, n]], {n, 140}]; tab

Prog

(PARI) isok(m) = kronecker(m, prime(m)) == 1; \\ Michel Marcus, Feb 06 2020

Crossrefs

Cf. A000040, A000290, A242750, A332021.

Sequence in context: A126004 A258251 A331086 * A184801 A228914 A200535

Adjacent sequences: A332017 A332018 A332019 * A332021 A332022 A332023

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2020

Status

approved