A107181 -id:A107181 - cp's OEIS frontend

A107132 Primes of the form 2x^2 + 13y^2.

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A140633 Primes of the form 7x^2+4xy+52y^2.

Original entry on oeis.org

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A229856 Primes of the form 384*k + 257.

Original entry on oeis.org

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041, 40193, 40577, 40961, 41729, 43649, 44417

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Every Fermat number greater than 257 has a prime factor of the form 384*k + 257, k > 0.

Wikipedia, Fermat number

Subsequence of A107181 (primes of the form 8x^2+9y^2).

Cf. A000215, A023394, A094358, A229853, A229855, A229856.

[384*n+257 : n in [0..115] | IsPrime(384*n+257)];

Select[Table[384*n + 257, {n, 0, 115}], PrimeQ]

A139527 Numbers n such that numbers 24n+5 are primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 12, 13, 16, 19, 21, 23, 27, 28, 29, 32, 33, 34, 39, 42, 44, 46, 49, 51, 53, 54, 57, 62, 67, 68, 71, 72, 78, 79, 81, 82, 83, 86, 89, 92, 93, 96, 97, 98, 99, 103, 106, 109, 112, 114, 116, 118, 119, 121, 123, 134, 141, 142, 144, 147, 148, 149, 153, 154

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Numbers n such that:
24n+1 is prime see A111174, primes 24n+1 see A107008
24n+5 is prime see A139527, primes 24n+5 see A107003
24n+7 is prime see A139483, primes 24n+7 see A107006
24n+11 is prime A139528, primes 24n+11 see A107007
24n+13 is prime see A139529, primes 24n+13 see A139530
24n+17 is prime see A139531, primes 24n+17 see A107181
24n+19 is prime see A139532, primes 24n+19 see A141373
24n+23 is prime see A131210, primes 24n+23 see A134517

Harvey P. Dale, Table of n, a(n) for n = 1..1000

a = {}; Do[If[PrimeQ[24 n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Table[(Prime[n]-5)/24,{n,800}],IntegerQ] (* Harvey P. Dale, Feb 25 2016 *)

A189242 Numbers n such that 24*n+17 is not prime.

Original entry on oeis.org

2, 6, 7, 8, 12, 13, 15, 17, 19, 20, 22, 27, 28, 29, 30, 32, 34, 37, 41, 42, 44, 46, 47, 48, 51, 52, 54, 55, 57, 60, 62, 63, 65, 67, 68, 69, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 87, 90, 91, 92, 93, 96, 97, 98, 102, 103, 104, 105, 106, 107, 111, 112, 117, 118

Offset: 1

Vincenzo Librandi, Apr 19 2011

			Distribution of the terms in the following triangular array:
*;
*,*;
*,*,*;
*,*,*,*;
*,*,*,*,*;
*,2,*,*,*,*;
*,*,*,*,*,*,*;
*,*,*,*,*,*,*,*;
*,*,*,*,8,*,*,*,*;
*,*,*,*,*,*,*,*,*,*;
*,*,6,*,*,*,*,*,*,*,*;
*,*,*,*,*,*,*,17,*,*,*,*; etc.
where * marks the non-integer values of (2*h*k + k + h - 8)/12 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A139531, A107181.

[n: n in [0..200] | not IsPrime(24*n+17)]; // Vincenzo Librandi, Apr 19 2011

Select[Range[1,300],!PrimeQ[24 # + 17] &] (* Vincenzo Librandi, Aug 05 2012 *)

A256397 Primes congruent to {17, 23} mod 24.

Original entry on oeis.org

17, 23, 41, 47, 71, 89, 113, 137, 167, 191, 233, 239, 257, 263, 281, 311, 353, 359, 383, 401, 431, 449, 479, 503, 521, 569, 593, 599, 617, 641, 647, 719, 743, 761, 809, 839, 857, 863, 881, 887, 911, 929, 953, 977, 983, 1031, 1049, 1097, 1103, 1151, 1193

Offset: 1

Arkadiusz Wesolowski, Jun 03 2015

nonn

All these primes do not divide any number of the form 2^k + 3. Therefore, they are not in A256396.

Cf. A107181, A134517, A256396.

[p: p in PrimesUpTo(1193) | p mod 24 in {17, 23}];

Select[Prime@Range[196], MemberQ[{17, 23}, Mod[#, 24]] &]

select(p->my(k=p%24); k==17||k==23, primes(1000)) \\ Charles R Greathouse IV, Jun 03 2015

A107181 UNION A134517.

A290402 Primes congruent to {7, 17} mod 24.

Original entry on oeis.org

7, 17, 31, 41, 79, 89, 103, 113, 127, 137, 151, 199, 223, 233, 257, 271, 281, 353, 367, 401, 439, 449, 463, 487, 521, 569, 593, 607, 617, 631, 641, 727, 751, 761, 809, 823, 857, 881, 919, 929, 953, 967, 977, 991, 1039, 1049, 1063, 1087, 1097, 1193, 1217, 1231

Offset: 1

Arkadiusz Wesolowski, Aug 03 2017

nonn

All these primes do not divide any number of the form 3*2^k - 1. Therefore, they are not in A001915.

Cf. A001915, A107006, A107181.

[p: p in PrimesUpTo(1231) | p mod 24 in {7, 17}];

Select[Prime@Range[202], MemberQ[{7, 17}, Mod[#, 24]] &]

A107006 UNION A107181.

cp's OEIS Frontend

A107132 Primes of the form 2x^2 + 13y^2.

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A140633 Primes of the form 7x^2+4xy+52y^2.

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A229856 Primes of the form 384*k + 257.

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A139527 Numbers n such that numbers 24n+5 are primes.

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A189242 Numbers n such that 24*n+17 is not prime.

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A256397 Primes congruent to {17, 23} mod 24.

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A290402 Primes congruent to {7, 17} mod 24.

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Formula