A105126 -id:A105126 - cp's OEIS frontend

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32y^2 and x^2 + 64y^2.

113, 257, 337, 353, 577, 593, 881, 1153, 1201, 1217, 1249, 1553, 1601, 1777, 1889, 2113, 2129, 2273, 2593, 2657, 2689, 2833, 3089, 3121, 3137, 3217, 3313, 3361, 3761, 4001, 4049, 4177, 4273, 4289, 4481, 4513, 4657, 4721, 4801, 4817, 4993, 5233, 5297, 5393

Offset: 1

Rémy Sigrist, Mar 27 2019

nonn

Kaplansky showed that prime numbers congruent to 1 modulo 16 are representable by both or neither of the quadratic forms x^2 + 32*y^2 and x^2 + 64*y^2. This sequence corresponds to those representable by both, and A325068 corresponds to those representable by neither.
Also, Kaplansky showed that prime numbers congruent to 9 modulo 16 are representable by exactly one of these quadratic forms. A325069 corresponds to those representable by the first form and A325070 to those representable by the second form.
Brink provided similar results for other congruences.

			Regarding 1201:
- 1201 is a prime number,
- 1201 = 75*16 + 1,
- 1201 = 7^2 + 32*6^2 = 25^2 + 64*3^2,
- hence 1201 belongs to the sequence.

David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, Journal of Number Theory 129(2) (2009), 464-468, doi:10.1016/j.jnt.2008.04.007, MR 2473893.
Rémy Sigrist, PARI program for A325067
Wikipedia, Kaplansky's theorem on quadratic forms

Cf. A094407, A105126, A325068, A325069, A325070.
See A325071, A325072, A325073 and A325074 for similar results in congruences modulo 16.
See A325075, A325076, A325077 and A325078 for similar results in congruences modulo 39.
See A325079, A325080, A325081 and A325082 for similar results in congruences modulo 55.
See A325083, A325084, A325085 and A325086 for similar results in congruences modulo 112.
See A325087, A325088, A325089 and A325090 for similar results in congruences modulo 240.

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A094407 Primes of the form 16n+1.

Original entry on oeis.org

17, 97, 113, 193, 241, 257, 337, 353, 401, 433, 449, 577, 593, 641, 673, 769, 881, 929, 977, 1009, 1153, 1201, 1217, 1249, 1297, 1361, 1409, 1489, 1553, 1601, 1697, 1777, 1873, 1889, 2017, 2081, 2113, 2129, 2161, 2273, 2417, 2593, 2609, 2657, 2689, 2753

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jun 03 2004

Subsequence of A007519 (primes of form 8n+1). - Zak Seidov, May 16 2012
Primes p such that p XOR 14 = p + 14. - Brad Clardy, Jul 23 2012
A prime of the form 16n+1 is represented either by both x^2+32y^2 and x^2+64y^2 or by neither (see Kaplansky link). - Michel Marcus, Dec 23 2012
Odd primes p such that -1 is an 8th power mod p. - Eric M. Schmidt, Mar 27 2014

T. D. Noe, Table of n, a(n) for n=1..1000
C, Caldwell, Prime test.
Irving Kaplansky, The forms x+32y^2 and x+64y^2, Proc. Amer. Math. Soc. 131 (2003), 2299-2300

Primes congruent to k mod 16: A094407, A091968, A127589, A141194, A105126, A141195, A141196, A127576.

Cf. A065091, A002144, A007519, A133870, A142925, A208177, A208178, A076339.

a094407 n = a094407_list !! (n-1)
a094407_list = filter ((== 1) . a010051) [1,17..]
-- Reinhard Zumkeller, Mar 06 2012

p:=proc(n) if isprime(16*n+1)=true then 16*n+1 else fi end:seq(p(n),n=1..200); # Emeric Deutsch, Dec 23 2004

lst={};Do[p=16*n+1;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[16*Range[200]+1,PrimeQ] (* Harvey P. Dale, Nov 04 2017 *)

More terms from Emeric Deutsch, Dec 23 2004

A105133 Numbers n such that 8n + 5 is prime.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 12, 13, 18, 19, 21, 22, 24, 28, 33, 34, 36, 39, 43, 46, 48, 49, 52, 57, 63, 67, 69, 76, 81, 82, 84, 87, 88, 91, 94, 96, 99, 102, 103, 106, 109, 117, 124, 126, 127, 132, 133, 136, 138, 139, 147, 151, 153, 154, 159, 162, 171, 172, 178, 181, 186, 193, 199, 201, 202

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105134-A105140, A002145, A007521, A105126-A105132.

[n: n in [0..500] | IsPrime(8*n+5)];  // Vincenzo Librandi, Jan 07 2013

M:=500; f:=proc(n) local t1,t2; t1:=[]; for k from 0 to M do t2:=2^n*k+2^(n-1)+1; if isprime(t2) then t1:=[op(t1),k]; fi; od; t1; end; f(3);

Select[Range[0, 300], PrimeQ[8 # + 5]&] (* Vincenzo Librandi, Jan 07 2013 *)

is(n)=isprime(8*n+5) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A007521(n)-5)/8. - Zak Seidov, Sep 08 2015

A105132 Primes of the form 1024n + 513.

Original entry on oeis.org

7681, 10753, 11777, 17921, 23041, 26113, 32257, 36353, 45569, 51713, 67073, 76289, 81409, 84481, 87553, 96769, 102913, 112129, 113153, 115201, 118273, 119297, 125441, 133633, 143873, 153089, 155137, 158209, 159233, 161281, 168449, 170497, 176641

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A002145, A007521, A105126-A105131.

[ a: n in [1..300] | IsPrime(a) where a is 1024*n+513 ]; // Vincenzo Librandi, Dec 07 2011

lst={};Do[p=1024*n+513;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[1024*Range[500]+513,PrimeQ] (* Harvey P. Dale, Jul 18 2011 *)

forprimestep(p=7681,176641,1024, print1(p", ")) \\ Charles R Greathouse IV, Nov 01 2022

a(n) ~ 512n log n. - Charles R Greathouse IV, Nov 01 2022

A105140 Numbers n such that 1024n+513 is prime.

Original entry on oeis.org

7, 10, 11, 17, 22, 25, 31, 35, 44, 50, 65, 74, 79, 82, 85, 94, 100, 109, 110, 112, 115, 116, 122, 130, 140, 149, 151, 154, 155, 157, 164, 166, 172, 179, 206, 211, 214, 215, 221, 227, 229, 232, 245, 254, 256, 259, 269, 271, 277, 280, 281, 292, 295, 296, 299, 316, 322, 332

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105133-A105139, A002145, A007521, A105126-A105132.

[n: n in [0..350] | IsPrime(1024*n + 513)]; // Vincenzo Librandi, Jan 08 2013

Select[Range[0, 500], PrimeQ[1024 # + 513]&] (* Vincenzo Librandi, Jan 08 2013 *)

is(n)=isprime(1024*n+513) \\ Charles R Greathouse IV, Feb 17 2017

A141194 Primes of the form 16k+7.

Original entry on oeis.org

7, 23, 71, 103, 151, 167, 199, 263, 311, 359, 439, 487, 503, 599, 631, 647, 727, 743, 823, 839, 887, 919, 967, 983, 1031, 1063, 1223, 1303, 1319, 1367, 1399, 1447, 1511, 1543, 1559, 1607, 1783, 1831, 1847, 1879, 2039, 2087, 2311, 2423, 2503, 2551, 2647

Offset: 1

T. D. Noe, Jun 12 2008

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A105126, A141195, A141196, A127576.

lst={};Do[If[PrimeQ[p=16*n+7],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[16*Range[0,200]+7,PrimeQ] (* Harvey P. Dale, Nov 26 2015 *)

A141195 Primes of the form 16k+11.

Original entry on oeis.org

11, 43, 59, 107, 139, 251, 283, 331, 347, 379, 443, 491, 523, 571, 587, 619, 683, 811, 827, 859, 907, 971, 1019, 1051, 1163, 1259, 1291, 1307, 1451, 1483, 1499, 1531, 1579, 1627, 1723, 1787, 1867, 1931, 1979, 2011, 2027, 2203, 2251, 2267, 2347, 2411, 2459

Offset: 1

T. D. Noe, Jun 12 2008

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A141194, A105126, A141196, A127576.

lst={};Do[If[PrimeQ[p=16*n+11],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[16*Range[0,200]+11,PrimeQ] (* Harvey P. Dale, Aug 22 2014 *)

A141196 Primes of the form 16k+13.

Original entry on oeis.org

13, 29, 61, 109, 157, 173, 269, 317, 349, 397, 461, 509, 541, 557, 653, 701, 733, 797, 829, 877, 941, 1021, 1069, 1117, 1181, 1213, 1229, 1277, 1373, 1453, 1549, 1597, 1613, 1693, 1709, 1741, 1789, 1901, 1933, 1949, 1997, 2029, 2141, 2221, 2237, 2269

Offset: 1

T. D. Noe, Jun 12 2008

nonn

Which sequence, this or A141194, produces more primes? The race is very close. For example, A141194(1000)=80599 and A141196(1000)=80909, a difference of just 32 primes after a thousand terms. - Art Baker, Aug 07 2019

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A094407, A091968, A127589, A141194, A105126, A141195, A127576.

lst={};Do[If[PrimeQ[p=16*n+13],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)

A105131 Primes of the form 512n+257.

Original entry on oeis.org

257, 769, 3329, 7937, 9473, 14081, 14593, 22273, 23297, 26881, 30977, 31489, 36097, 37633, 40193, 41729, 43777, 46337, 49409, 49921, 57089, 57601, 60161, 70913, 75521, 77569, 78593, 84737, 88321, 91393, 96001, 98561, 100609, 103681, 106753, 107777

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A002145, A007521, A105126-A105132.

[ a: n in [0..210] | IsPrime(a) where a is 512*n+257 ]; // Vincenzo Librandi, Jul 19 2012

Select[Table[512*n+257,{n,0,800}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

A105127 Primes of the form 32n+17.

Original entry on oeis.org

17, 113, 241, 337, 401, 433, 593, 881, 977, 1009, 1201, 1297, 1361, 1489, 1553, 1777, 1873, 2129, 2161, 2417, 2609, 2801, 2833, 2897, 3089, 3121, 3217, 3313, 3697, 3761, 3793, 3889, 4049, 4177, 4241, 4273, 4337, 4561, 4657, 4721, 4817, 5009, 5233, 5297

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A002145, A007521, A105126-A105132.

[ a: n in [0..200] | IsPrime(a) where a is 32*n+17 ]; // Vincenzo Librandi, Jul 19 2012

Select[Range[17,20000,32],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
Select[Table[32*n+17,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

cp's OEIS Frontend

A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32*y^2 and x^2 + 64*y^2.

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PARI

A094407 Primes of the form 16n+1.

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A105133 Numbers n such that 8n + 5 is prime.

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Magma

Maple

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PARI

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A105132 Primes of the form 1024n + 513.

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Magma

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A105140 Numbers n such that 1024n+513 is prime.

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Magma

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PARI

A141194 Primes of the form 16k+7.

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Mathematica

A141195 Primes of the form 16k+11.

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Mathematica

A141196 Primes of the form 16k+13.

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A325067 Prime numbers congruent to 1 modulo 16 representable by both x^2 + 32y^2 and x^2 + 64y^2.