A091968 -id:A091968 - cp's OEIS frontend

A094407 Primes of the form 16n+1.

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

A092022 Numbers k such that 16k + 3 is prime.

Original entry on oeis.org

0, 1, 4, 5, 8, 10, 11, 13, 14, 19, 26, 29, 31, 34, 35, 40, 41, 43, 46, 49, 55, 59, 68, 70, 73, 74, 80, 89, 91, 95, 98, 101, 104, 106, 109, 113, 119, 124, 125, 130, 131, 133, 136, 140, 146, 148, 154, 158, 161, 166, 169, 175, 176, 178, 185, 188, 199, 200, 203, 206, 208

Offset: 1

Ray Chandler, Mar 15 2004

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A091968, A093012.

[n: n in [0..300] |IsPrime(16*n+3)]; // Vincenzo Librandi, Mar 28 2015

Select[Range[0, 208], PrimeQ[16# + 3] &] (* Ray Chandler, Mar 15 2004 *)

is(n)=isprime(16*n+3) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A091968(n)-3)/16. - Zak Seidov, Mar 28 2015

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 *)

A093012 Numbers k such that prime(k) == 3 (mod 16).

Original entry on oeis.org

2, 8, 19, 23, 32, 38, 41, 47, 49, 63, 81, 91, 95, 101, 103, 117, 120, 125, 131, 138, 153, 161, 182, 188, 193, 195, 208, 225, 232, 241, 248, 256, 262, 266, 272, 280, 292, 300, 304, 314, 317, 321, 327, 334, 346, 351, 365, 370, 376, 385, 394, 409, 410, 414, 427

Offset: 1

Ray Chandler, Mar 15 2004, revised Nov 06 2006

nonn

A091968 indexed by A000040.

The asymptotic density of this sequence is 1/8 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000720, A091968, A092022.

[n: n in [1..450]|(NthPrime(n) mod 16) eq 3] // G. C. Greubel, Feb 05 2019

Select[Range[430], Mod[Prime[ # ], 16] == 3 &] (* Ray Chandler, Nov 06 2006 *)

{isok(n) = Mod(prime(n), 16)==3};
for(n=1, 450, if(isok(n)==1, print1(n, ", "), 0)) \\ G. C. Greubel, Feb 05 2019

A000040(a(n)) = A091968(n).

a(n) = A000720(A091968(n)).

A228228 Primes congruent to {3, 5, 13, 15} mod 16.

Original entry on oeis.org

3, 5, 13, 19, 29, 31, 37, 47, 53, 61, 67, 79, 83, 101, 109, 127, 131, 149, 157, 163, 173, 179, 181, 191, 197, 211, 223, 227, 229, 239, 269, 271, 277, 293, 307, 317, 349, 367, 373, 383, 389, 397, 419, 421, 431, 461, 463, 467, 479, 499, 509, 541, 547, 557, 563

Offset: 1

Arkadiusz Wesolowski, Aug 16 2013

nonn

Union of A091968, A127589, A141196, and A127576.

Let p be a prime number and let E(p) denote the elliptic curve y^2 = x^3 + p*x. If p is in the sequence, then the rank of E(p) is 0 or 1. Therefore, A060953(a(n)) must be one of only two values: 0 or 1.

J. H. Silverman, The arithmetic of elliptic curves, Springer, NY, 1986, p. 311.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A007519, A060953, A091968, A127576, A127589, A141196, A228227.

[p: p in PrimesUpTo(563) | p mod 16 in {3, 5, 13, 15}];

Select[Prime@Range[103], MemberQ[{3, 5, 13, 15}, Mod[#, 16]] &]

A321212 Numbers that are congruent to {2, 3} mod 16.

Original entry on oeis.org

2, 3, 18, 19, 34, 35, 50, 51, 66, 67, 82, 83, 98, 99, 114, 115, 130, 131, 146, 147, 162, 163, 178, 179, 194, 195, 210, 211, 226, 227, 242, 243, 258, 259, 274, 275, 290, 291, 306, 307, 322, 323, 338, 339, 354, 355, 370, 371, 386, 387, 402, 403, 418, 419, 434, 435, 450, 451, 466, 467, 482, 483, 498, 499

Offset: 1

Yuen Biu, Oct 31 2018

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047608, A151977.

A091968 is a subsequence.

[n: n in [1..500] | n mod 16 in [2, 3]]; // Vincenzo Librandi, Nov 30 2018

Flatten@ Array[16 # + {2, 3} &, 31, 0] (* Michael De Vlieger, Oct 31 2018 *)
Select[Range[1, 500], MemberQ[{2,3}, Mod[#, 16]] &] (* Vincenzo Librandi, Nov 30 2018 *)

a(n) = (16*n - 7*(-1)^n - 19)/2 \\ David Lovler, Aug 20 2022

a(n) = A151977(n) + 2.
G.f.: x*(2 + x + 13*x^2)/((-1 + x)^2*(1 + x)). - Stefano Spezia, Nov 01 2018
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3. - Chai Wah Wu, Nov 29 2018
From Franck Maminirina Ramaharo, Nov 30 2018: (Start)
a(n) = (16*n - 7*(-1)^n - 19)/2.
E.g.f.: (-7 + 26*exp(x) - 19*exp(2*x) + 16*x*exp(2*x))/(2*exp(x)). (End)
E.g.f.: 13 + ((16*x -19)*exp(x) - 7*exp(-x))/2. - David Lovler, Aug 20 2022
a(n) = a(n-2) + 16. - David A. Corneth, Nov 30 2018

cp's OEIS Frontend

A094407 Primes of the form 16n+1.

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A092022 Numbers k such that 16k + 3 is prime.

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A141194 Primes of the form 16k+7.

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A141195 Primes of the form 16k+11.

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A141196 Primes of the form 16k+13.

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A093012 Numbers k such that prime(k) == 3 (mod 16).

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A228228 Primes congruent to {3, 5, 13, 15} mod 16.

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Magma

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A321212 Numbers that are congruent to {2, 3} mod 16.

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Magma