A296920 -id:A296920 - cp's OEIS frontend

A296935 Rational primes that decompose in the field Q(sqrt(11)).

5, 7, 19, 37, 43, 53, 79, 83, 89, 97, 107, 113, 127, 131, 137, 139, 151, 157, 167, 181, 211, 227, 229, 239, 257, 263, 269, 271, 283, 307, 313, 317, 347, 353, 359, 389, 397, 401, 421, 431, 433, 439, 449, 479, 491, 503, 509, 521, 523, 547, 563, 571, 577, 607

Offset: 1

N. J. A. Sloane, Dec 26 2017

Cf. A296936, A038881, A038882.

Load the Maple program HH given in A296920. Then run HH(11, 200); This produces A296935, A296936, A038881, A038882.

Select[Prime[Range[120]], KroneckerSymbol[11, #] == 1 &] (* Amiram Eldar, Dec 24 2023 *)

list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(11,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

A296937 Rational primes that decompose in the field Q(sqrt(13)).

Original entry on oeis.org

3, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

N. J. A. Sloane, Dec 26 2017

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018
From Jianing Song, Apr 21 2022: (Start)
Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.
Primes p such that p^6 == 1 (mod 13).
Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

Cf. A011583 (kronecker symbol modulo 13), A038883.
Rational primes that decompose in the quadratic field with discriminant D: A139513 (D=-20), A191019 (D=-19), A191018 (D=-15), A296920 (D=-11), A033200 (D=-8), A045386 (D=-7), A002144 (D=-4), A002476 (D=-3), A045468 (D=5), A001132 (D=8), A097933 (D=12), this sequence (D=13), A296938 (D=17).
Cf. A038884 (inert rational primes in the field Q(sqrt(13))).

Load the Maple program HH given in A296920. Then run HH(13, 200); This produces A296937, A038884, A038883.

isA296937(p) = isprime(p) && kronecker(p,13) == 1 \\ Jianing Song, Apr 21 2022

Equals A038883 \ {13}. - Jianing Song, Apr 21 2022

A155488 Primes p with property that p^2 is of the form x^2 + 40y^2.

Original entry on oeis.org

7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, 157, 167, 173, 179, 197, 211, 223, 241, 251, 263, 277, 281, 293, 317, 331, 367, 373, 379, 383, 397, 401, 409, 419, 449, 463, 487, 491, 499, 503, 521, 557, 569, 571, 601, 607, 613, 619, 641, 647

Offset: 1

Zak Seidov, Jan 23 2009

nonn

All p^2 are congruent to {1, 9} (mod 40), as in A107145.

Rational primes that decompose in the field Q(sqrt(-10)). - N. J. A. Sloane, Dec 26 2017

Robert Price, Table of n, a(n) for n = 1..1000
Zak Seidov, First 1000 terms
Index to sequences related to decomposition of primes in quadratic fields

Cf. A107145 (Primes of the form x^2 + 40y^2).

Load the Maple program HH given in A296920. Then run HH(-10, 200); This produces A155488, A296925, A293859. - N. J. A. Sloane, Dec 26 2017

Sqrt[#]&/@Select[Union[Flatten[Table[x^2+40y^2,{x,800},{y,100}]]],PrimeQ[Sqrt[#]]&] (* Harvey P. Dale, Jul 12 2025 *)

A293859 Prime factors of numbers of the form k^2 + 10.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, 157, 167, 173, 179, 197, 211, 223, 241, 251, 263, 277, 281, 293, 317, 331, 367, 373, 379, 383, 397, 401, 409, 419, 449, 463, 487, 491, 499, 503, 521, 557, 569, 571, 601, 607, 613, 619, 641

Offset: 1

J. Lowell, Oct 17 2017

nonn

Primes p such that Legendre(-10,p) = 0 or 1. - N. J. A. Sloane, Dec 26 2017
Question: Is there a comment of the form "a prime number is in this sequence if and only if it is congruent to (list of appropriate values) mod n" for this sequence?
From Robert Israel, Nov 19 2017: (Start)
Prime p > 5 is in the sequence iff -10 is a quadratic residue mod p.
Thus p is either in the intersection of A002144 and A038879 or in neither of them.
Primes == 1, 2, 5, 7, 9, 11, 13, 19, 23, or 37 (mod 40). (End)

			7 is in the sequence because 2^2 + 10 = 14 is 2 times 7.
19 is in the sequence because 3^2 + 10 = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002144, A038879, A114948.

select(isprime, [seq(seq(i*40+j, j = [1, 2, 5, 7, 9, 11, 13, 19, 23, 37]), i=0..40)]); # Robert Israel, Nov 19 2017
# Load the Maple program HH given in A296920. Then run HH(-10, 200); This produces A155488, A296925, A293859. - N. J. A. Sloane, Dec 26 2017

Select[Prime@ Range@ 120, {} != FindInstance[# x == n^2 + 10 && n >= 0 && x > 0, {n, x}, Integers, 1] &] (* Giovanni Resta, Oct 19 2017 *)

More terms from Giovanni Resta, Oct 19 2017

A296915 Primes that are not squares mod 163.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 59, 67, 73, 79, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 181, 191, 193, 211, 229, 233, 239, 241, 257, 269, 271, 277, 283, 293, 311, 317, 331, 337, 349, 353, 389, 401, 431, 433, 443, 449, 463, 467, 479, 491, 509, 521, 541

Offset: 1

Ed Pegg Jr, Dec 22 2017

nonn

Inert rational primes in Q(sqrt -163). (Note that 41 is not inert in this field, it decomposes - see A296921.)

Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.

Load the Maple program HH given in A296920. Then run HH(-163,200); - N. J. A. Sloane, Dec 26 2017

lista(nn) = forprime(p=2, nn, if (!issquare(Mod(p, 163)), print1(p, ", "));); \\ Michel Marcus, Dec 24 2017

Corrected by N. J. A. Sloane, Dec 25 2017 (including deletion of incorrect comments in CROSS-REFERENCES)

A296921 Rational primes that decompose in the field Q(sqrt(-163)).

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 167, 173, 179, 197, 199, 223, 227, 251, 263, 281, 307, 313, 347, 359, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 547, 563, 577, 593, 607, 641, 647, 653, 661, 673, 677, 691, 701, 709, 733, 739, 743, 773, 787, 797

Offset: 1

N. J. A. Sloane, Dec 25 2017

From Jianing Song, Oct 13 2022: (Start)
Primes p such that kronecker(-163,p) = 1 (or equivalently, kronecker(p,163) = 1).
Primes p such that p^81 == 1 (mod 163). (End)

A257362, the sequence of primes that do not remain inert in the field Q(sqrt(-163)), is essentially the same.

Cf. A296915 (rational primes that remain inert in the field Q(sqrt(-163))).

Load the Maple program HH given in A296920. Then run HH(-163,200);

isA296921(p) = isprime(p) && kronecker(p,163) == 1

A296922 Primes p such that Legendre(-5,p) = 0 or 1.

Original entry on oeis.org

3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563, 569, 587, 601, 607, 641, 643

Offset: 1

N. J. A. Sloane, Dec 25 2017

nonn

Primes == 1, 3, 5, 7, or 9 (mod 20). Primes whose 10's digit is even. - Robert Israel, Dec 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A139513, A240920, A296920, A296923.

Load the Maple program HH given in A296920. Then run HH(-5,200);
select(isprime, {seq(seq(20*i+j,j=[1,3,5,7,9]),i=0..100)}); # Robert Israel, Dec 27 2017

Select[Prime@ Range@ 120, MemberQ[{0, 1}, KroneckerSymbol[-5, #]] &] (* or *)
Select[Prime@ Range@ 120, MemberQ[Range[1, 9, 2], Mod[#, 20]] &] (* Michael De Vlieger, Jan 02 2018 *)

lista(nn) = forprime(p=2, nn, if (kronecker(-5,p) >= 0, print1(p, ", "))); \\ Michel Marcus, Dec 26 2017

a(n) = A240920(n+1) for n >= 1. - Georg Fischer, Oct 30 2018

A296923 Primes p such that Legendre(-5,p) = -1.

Original entry on oeis.org

2, 11, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 211, 233, 239, 251, 257, 271, 277, 293, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 419, 431, 433, 439, 457, 479, 491, 499, 557, 571, 577, 593, 599, 613

Offset: 1

N. J. A. Sloane, Dec 25 2017

nonn

Primes == 2, 11, 13, 17, or 19 (mod 20). - Robert Israel, Dec 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A296922, A003626.

Load the Maple program HH given in A296920. Then run HH(-5,200);
select(isprime, {seq(seq(20*i+j,j=[2,11,13,17,19]),i=0..100)}); # Robert Israel, Dec 27 2017

lista(nn) = forprime(p=2, nn, if (kronecker(-5,p) == -1, print1(p, ", "))); \\ Michel Marcus, Dec 26 2017

A296929 Rational primes that decompose in the field Q(sqrt(-17)).

Original entry on oeis.org

3, 7, 11, 13, 23, 31, 53, 71, 79, 89, 101, 107, 131, 137, 139, 149, 157, 163, 167, 199, 211, 227, 229, 257, 281, 283, 293, 311, 347, 349, 353, 367, 373, 379, 389, 409, 419, 421, 431, 433, 439, 457, 461, 479, 487, 499, 503, 509, 547, 557, 569, 571, 577, 593, 607

Offset: 1

N. J. A. Sloane, Dec 26 2017

Load the Maple program HH given in A296920. Then run HH(-17, 200); This produces A296929, A296930, A296931.

list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(-17,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

Primes == {1, 3, 7, 9, 11, 13, 21, 23, 25, 27, 31, 33, 39, 49, 53, 63} (mod 68). - Travis Scott, Jan 05 2023

A296931 Primes p such that Legendre(-17,p) = 0 or 1.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 23, 31, 53, 71, 79, 89, 101, 107, 131, 137, 139, 149, 157, 163, 167, 199, 211, 227, 229, 257, 281, 283, 293, 311, 347, 349, 353, 367, 373, 379, 389, 409, 419, 421, 431, 433, 439, 457, 461, 479, 487, 499, 503, 509, 547, 557

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Primes p such that p == 1, 2, 3, 7, 9, 11, 13, 17, 21, 23, 25, 27, 31, 33, 39, 49, 53, or 63 (mod 68). - Robert Israel, Dec 26 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Load the Maple program HH given in A296920. Then run HH(-17, 200); This produces A296929, A296930, A296931.
Alternative:
select(p-> isprime(p) and numtheory:-legendre(-17,p)<>-1, [2,seq(i,i=3..1000)]); # Robert Israel, Dec 26 2017

cp's OEIS Frontend

A296935 Rational primes that decompose in the field Q(sqrt(11)).

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A296937 Rational primes that decompose in the field Q(sqrt(13)).

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A155488 Primes p with property that p^2 is of the form x^2 + 40y^2.

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A293859 Prime factors of numbers of the form k^2 + 10.

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A296915 Primes that are not squares mod 163.

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A296921 Rational primes that decompose in the field Q(sqrt(-163)).

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A296922 Primes p such that Legendre(-5,p) = 0 or 1.

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A296923 Primes p such that Legendre(-5,p) = -1.

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