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A332020
Positive integers m which are quadratic residues modulo prime(m).
2
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2020
STATUS
approved