A127589 -id:A127589 - 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

A127590 Numbers n such that 16n+5 is prime.

Original entry on oeis.org

0, 2, 3, 6, 9, 11, 12, 14, 17, 18, 23, 24, 26, 38, 41, 42, 44, 47, 48, 51, 53, 62, 63, 66, 68, 69, 77, 81, 86, 89, 93, 101, 102, 104, 108, 116, 117, 123, 128, 129, 138, 143, 144, 146, 147, 149, 152, 159, 167, 168, 171, 174, 177, 182, 191, 194

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589.

a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[0,200],PrimeQ[16#+5]&] (* Harvey P. Dale, Aug 31 2020 *)

is(n)=isprime(16*n+5) \\ Charles R Greathouse IV, Feb 17 2017

A127591 Numbers k such that 64k+21 is prime.

Original entry on oeis.org

2, 4, 10, 13, 17, 19, 20, 22, 23, 25, 29, 32, 37, 44, 50, 53, 55, 58, 59, 62, 68, 79, 83, 88, 89, 94, 95, 97, 100, 107, 109, 113, 118, 122, 134, 142, 143, 152, 155, 157, 158, 163, 167, 169, 173, 193, 194, 199, 200

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

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

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127592, A127593, A127594.

a = {}; Do[If[PrimeQ[21 + 64 n], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[200],PrimeQ[64#+21]&] (* Harvey P. Dale, Jan 15 2016 *)

A127592 Primes of the form 64k+21.

Original entry on oeis.org

149, 277, 661, 853, 1109, 1237, 1301, 1429, 1493, 1621, 1877, 2069, 2389, 2837, 3221, 3413, 3541, 3733, 3797, 3989, 4373, 5077, 5333, 5653, 5717, 6037, 6101, 6229, 6421, 6869, 6997, 7253, 7573, 7829, 8597, 9109, 9173, 9749, 9941, 10069, 10133, 10453

Offset: 1

Artur Jasinski, Jan 19 2007, Nov 12 2007

All these primes are sums of two squares, also all indices are sums of two squares since we have the identity 64k+21 = 4(4(4k+1)+1)+1.

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

Cf. A000040. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127593, A127594.

[p: p in PrimesUpTo(11000) | p mod 64 eq 21 ]; // Vincenzo Librandi, Sep 06 2012

a = {}; Do[If[PrimeQ[21 + 64 n], AppendTo[a, 21 + 64 n]], {n, 0, 200}]; a
Select[Prime[Range[1700]], MemberQ[{21}, Mod[#, 64]] &] (* Vincenzo Librandi, Sep 06 2012 *)

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 *)

A127593 Primes of the form 256 k + 85.

Original entry on oeis.org

853, 1109, 1621, 1877, 2389, 3413, 5717, 6229, 6997, 7253, 10069, 10837, 11093, 12373, 13397, 16981, 17749, 18517, 18773, 19541, 21589, 22613, 23893, 24917, 27733, 29269, 30293, 31573, 32341, 37717, 39509, 40277, 41813, 43093, 46933

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127594.

a = {}; Do[If[PrimeQ[85 + 256 n], AppendTo[a, 85 + 256 n]], {n, 0, 200}]; a
Select[256*Range[200]+85,PrimeQ] (* Harvey P. Dale, Oct 09 2020 *)

A127594 Numbers k such that 256 k + 85 is prime.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 22, 24, 27, 28, 39, 42, 43, 48, 52, 66, 69, 72, 73, 76, 84, 88, 93, 97, 108, 114, 118, 123, 126, 147, 154, 157, 163, 168, 183, 184, 186, 196, 198

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593.

a = {}; Do[If[PrimeQ[85 + 256 n], AppendTo[a, n]], {n, 0, 200}]; a

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 *)

A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.

Original entry on oeis.org

2, 1, 0, 2, 3, 2, 4, 4, 3, 10, 3, 3, 2, 7, 2, 25, 6, 17, 4, 13, 3, 20, 36, 20, 11, 27, 66, 23, 39, 24, 19, 13, 3, 10, 6, 122, 71, 58, 24, 13, 3, 2, 41, 10, 6, 32, 58, 17, 4, 79, 26, 55, 36, 48, 31, 28, 9, 2, 76, 24, 32, 28, 63, 20, 37, 9, 2, 7, 39, 10, 91, 47

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..2391

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593, A127594, A127598.

a = {}; Do[k = 0; While[ !PrimeQ[k 4^n + (4^n - 1)/3], k++ ]; AppendTo[a, k], {n, 0, 50}]; a (*Artur Jasinski*)
lnk[n_]:=Module[{k=0,n4=4^n},While[!PrimeQ[k*n4+(n4-1)/3],k++];k]; Array[ lnk,60,0] (* Harvey P. Dale, May 28 2018 *)

from sympy import isprime
def a(n):
    k, fourn = 0, 4**n
    while not isprime(k*fourn + (fourn-1)//3): k += 1
    return k
print([a(n) for n in range(72)]) # Michael S. Branicky, May 18 2022

Offset corrected and a(51) and beyond from Michael S. Branicky, May 18 2022

cp's OEIS Frontend

A094407 Primes of the form 16n+1.

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A127590 Numbers n such that 16n+5 is prime.

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A127591 Numbers k such that 64k+21 is prime.

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A127592 Primes of the form 64k+21.

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A141194 Primes of the form 16k+7.

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A127593 Primes of the form 256 k + 85.

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A127594 Numbers k such that 256 k + 85 is prime.

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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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A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.