A191018 -id:A191018 - cp's OEIS frontend

A033212 Primes congruent to 1 or 19 (mod 30).

19, 31, 61, 79, 109, 139, 151, 181, 199, 211, 229, 241, 271, 331, 349, 379, 409, 421, 439, 499, 541, 571, 601, 619, 631, 661, 691, 709, 739, 751, 769, 811, 829, 859, 919, 991, 1009, 1021, 1039, 1051, 1069, 1129, 1171, 1201, 1231, 1249, 1279, 1291, 1321, 1381

Offset: 1

N. J. A. Sloane

Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.
Equivalently, primes congruent to 1 or 4 (mod 15). Also x^2+xy+4y^2 is the principal form of (fundamental) discriminant -15. The only other class for -15 contains the form 2x^2+xy+2y^2 (A106859), in the other genus. - Rick L. Shepherd, Jul 25 2014
Three further theorems (these were originally stated as conjectures, but are now known to be theorems, thanks to the work of J. B. Tunnell - see link):
1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative. - T. D. Noe, Apr 29 2008
2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative. - T. D. Noe, Apr 29 2008
3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - N. J. A. Sloane, Jun 01 2014
Also primes of the form x^2+7*x*y+y^2 (discriminant 45).
Lemma (Will Jagy, Jun 12 2014): If c is any (positive or negative) even number, then x^2 + x y + c y^2 and x^2 + (4 c - 1) y^2 represent the same odd numbers.
Proof: x (x + y) + c y^2 = odd, therefore x is odd,  x + y odd, so y is even. Let y = 2 t. Then x( x + 2 t) + 4 c t^2 = x^2 + 2 x t  + 4 c t^2 = (x+t)^2 + (4c-1) t^2 = odd. QED With c = 4, neither one represents 2, so x^2+15y^2 and x^2+xy+4y^2 represent the same primes.
Also, primes which are squares (mod 3*5). Subsequence of A191018. - David Broadhurst and M. F. Hasler, Jan 15 2016

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A106856, A139643, A106859.
Primes in A243173 and in A243174.
Cf. A141785 (d=45), A033212 (Primes of form x^2+15*y^2), A038872(d=5), A038873 (d=8), A068228, A141123 (d=12), A038883 (d=13), A038889 (d=17), A141111, A141112 (d=65).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[1, 0, 15, 10000] (* see A106856 *)
Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

select(n->n%30==1||n%30==19, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

is(p)=issquare(Mod(p,15))&&isprime(p) \\ M. F. Hasler, Jan 15 2016

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

Edited by N. J. A. Sloane, Jun 01 2014 and Oct 18 2014: added Tunnell document, revised entry, merged with A141184. The latter entry was submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008.

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A191017 Primes with Kronecker symbol (p|14) = 1.

Original entry on oeis.org

3, 5, 13, 19, 23, 59, 61, 71, 79, 83, 101, 113, 127, 131, 137, 139, 151, 157, 173, 181, 191, 193, 227, 229, 233, 239, 251, 263, 269, 281, 283, 293, 307, 337, 349, 359, 397, 401, 419, 431, 449, 457, 461, 463, 467, 487, 509, 523, 563, 569, 587, 599, 617, 619

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "Primes that are squares mod 14", which is sequence A045373. - M. F. Hasler, Jan 15 2016
Conjecture: primes congruent to {1, 3, 5, 9, 13, 15, 19, 23, 25, 27, 39, 45} mod 56. - Vincenzo Librandi, Jun 22 2016
From Jianing Song, Nov 21 2019: (Start)
Rational primes that decompose in the field Q(sqrt(-14)).
These are primes p such that either one of (a) p == 1, 2, 4 (mod 7), p == 1, 7 (mod 8) or (b) p == 3, 5, 6 (mod 7), p == 3, 5 (mod 8) holds. So the conjecture above is correct. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191018, A191020, A191061, A274504.

[p: p in PrimesUpTo(619) | KroneckerSymbol(p, 14) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,14]==1&]

is(p)=kronecker(p, 14)==1&&isprime(p) \\ Michel Marcus, Jun 23 2016 after A191032

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191020 Primes p with Kronecker symbol (p|2*11) = 1.

Original entry on oeis.org

13, 19, 23, 29, 31, 43, 47, 61, 71, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 173, 191, 197, 199, 211, 223, 227, 257, 277, 283, 293, 307, 311, 313, 347, 349, 353, 367, 373, 383, 401, 433, 449, 461, 463, 487, 491, 521, 523, 541, 547, 557, 563

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 22)", which is sequence A056874. - M. F. Hasler, Jan 15 2016

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A191017, A191018, A191023.

[p: p in PrimesUpTo(563) | KroneckerSymbol(p, 22) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,22]==1&]

lista(nn) = forprime(p=13, nn, if(kronecker(p, 22)==1, print1(p, ", "))) \\ Iain Fox, Mar 05 2018

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191023 Primes p which have Kronecker symbol (p|30) = 1.

Original entry on oeis.org

11, 13, 17, 23, 29, 31, 37, 43, 47, 59, 67, 79, 101, 113, 131, 137, 149, 151, 157, 163, 167, 179, 199, 233, 241, 251, 257, 263, 269, 271, 277, 283, 307, 353, 373, 383, 389, 397, 409, 419, 439, 461, 491, 503, 509, 523, 547, 593, 601, 613, 617, 631, 643, 647

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 30)", which is sequence A033212. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020.

[p: p in PrimesUpTo(647) | KroneckerSymbol(p, 30) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,30]==1&]

is(n)=isprime(n) && kronecker(n,30)==1 \\ Charles R Greathouse IV, Jul 12 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191025 Primes p which have Kronecker symbol (p|34) = 1.

Original entry on oeis.org

3, 5, 11, 29, 37, 47, 61, 89, 103, 107, 109, 127, 131, 137, 139, 151, 163, 173, 181, 191, 197, 211, 223, 227, 239, 257, 263, 269, 271, 277, 281, 283, 317, 347, 353, 359, 379, 383, 397, 409, 419, 433, 457, 463, 499, 541, 547, 569, 571, 577, 593, 599, 619, 631

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 34)", which is sequence A038889. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023.

[p: p in PrimesUpTo(631) | KroneckerSymbol(p, 34) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,34]==1&]

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191026 Primes p that have Jacobi symbol (p|35) = 1.

Original entry on oeis.org

3, 11, 13, 17, 29, 47, 71, 73, 79, 83, 97, 103, 109, 149, 151, 157, 167, 173, 179, 191, 211, 223, 227, 239, 257, 281, 283, 293, 307, 313, 331, 353, 359, 367, 379, 383, 389, 397, 401, 421, 431, 433, 449, 467, 491, 499, 503, 523, 541, 563, 569, 571, 577, 587

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "Primes which are squares (mod 35)", which is subsequence A106881. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025.

[p: p in PrimesUpTo(587) | JacobiSymbol(p,35) eq 1]; // Vincenzo Librandi, Sep 10 2012

Select[Prime[Range[200]], JacobiSymbol[#,35]==1&]

is(p)=kronecker(p, 35)==1&&isprime(p) \\ M. F. Hasler, Jan 15 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191028 Primes p with Kronecker symbol (p|38) = 1.

Original entry on oeis.org

3, 7, 13, 17, 23, 29, 37, 47, 53, 59, 67, 73, 107, 109, 137, 173, 179, 181, 191, 199, 211, 227, 233, 239, 263, 269, 271, 293, 307, 311, 313, 317, 331, 353, 359, 367, 373, 379, 421, 457, 463, 479, 503, 509, 523, 547, 563, 577, 593, 617, 631, 647, 659, 661

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 38)", which is sequence A106863. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025, A191026.

[p: p in PrimesUpTo(661) | KroneckerSymbol(p, 38) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,38]==1&]

is(p)=kronecker(p, 38)==1&&isprime(p) \\ M. F. Hasler, Jan 15 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191029 Primes p with Jacobi symbol (p|39) = 1.

Original entry on oeis.org

2, 5, 11, 41, 43, 47, 59, 61, 71, 79, 83, 89, 103, 127, 137, 139, 149, 157, 167, 181, 197, 199, 211, 227, 239, 277, 281, 283, 293, 313, 317, 337, 353, 359, 367, 373, 383, 401, 431, 433, 439, 449, 461, 479, 509, 523, 547, 557, 571, 587, 593, 601, 607, 617

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 39)", which is subsequence A267455 \ {3, 13}. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028.

[p: p in PrimesUpTo(617) | JacobiSymbol(p, 39) eq 1]; // Vincenzo Librandi, Sep 10 2012

Select[Prime[Range[200]], JacobiSymbol[#,39]==1&]

select(p->kronecker(p,39)==1&&isprime(p),[1..1000]) \\ M. F. Hasler, Jan 15 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191032 Primes p with Kronecker symbol (p|46) = 1.

Original entry on oeis.org

5, 11, 19, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 107, 109, 127, 149, 151, 157, 167, 181, 193, 223, 227, 229, 233, 239, 251, 257, 271, 283, 293, 311, 353, 373, 379, 389, 409, 419, 421, 439, 449, 463, 467, 487, 523, 557, 563, 571, 577, 593, 599, 601, 607

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 46)". - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028, A191029.

[p: p in PrimesUpTo(607) | KroneckerSymbol(p, 46) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,46]==1&]

select(p->kronecker(p,46)==1&&isprime(p),[1..1000]) \\ M. F. Hasler, Jan 15 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A191036 Primes p that have Jacobi symbol (p|55) = 1.

Original entry on oeis.org

2, 7, 13, 17, 31, 43, 59, 71, 73, 83, 89, 107, 127, 167, 173, 179, 181, 191, 193, 197, 199, 227, 229, 233, 251, 263, 269, 277, 283, 293, 307, 311, 331, 337, 347, 373, 379, 389, 401, 419, 421, 449, 457, 499, 503, 509, 521, 523, 547, 557, 563, 593, 599, 607

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares mod 55", which is sequence A267478, a subsequence whose terms have (p|5) = (p|11) = 1 except for the two initial terms 5 and 11. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023, A191025, A191026, A191028, A191029,A191032, A191034, A191037.

[p: p in PrimesUpTo(607) | JacobiSymbol(p, 55) eq 1]; // Vincenzo Librandi, Sep 10 2012

Select[Prime[Range[200]], JacobiSymbol[#,55]==1&]

select(p->kronecker(p,55)==1&&isprime(p),[1..1500]) \\ M. F. Hasler, Jan 15 2016

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

cp's OEIS Frontend

A033212 Primes congruent to 1 or 19 (mod 30).

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A191017 Primes with Kronecker symbol (p|14) = 1.

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A191020 Primes p with Kronecker symbol (p|2*11) = 1.

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A191023 Primes p which have Kronecker symbol (p|30) = 1.

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A191025 Primes p which have Kronecker symbol (p|34) = 1.

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A191026 Primes p that have Jacobi symbol (p|35) = 1.

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A191028 Primes p with Kronecker symbol (p|38) = 1.

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