A007521 -id:A007521 - cp's OEIS frontend

A007528 Primes of the form 6k-1.

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
From Bernard Schott, Feb 14 2019: (Start)
A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A004770 Numbers of the form 8k+5; or, numbers whose binary expansion ends in 101.

Original entry on oeis.org

5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109, 117, 125, 133, 141, 149, 157, 165, 173, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 389, 397, 405, 413, 421, 429, 437, 445

Offset: 1

N. J. A. Sloane

Only numbers of the form 8k+5 may be written as a sum of 5 odd squares. Examples: 5 = 1+1+1+1+1, 13 = 9+1+1+1+1, 21 = 9+9+1+1+1, 29 = 25+1+1+1+1= 9+9+9+1+1, 37 = 9+9+9+9+1 = 25+9+1+1+1, 45 = 25+9+9+1+1=9+9+9+9+9, 53 = 49+1+1+1+1 = 25+25+1+1+1 = 25+9+9+9+1, ... - Philippe Deléham, Sep 03 2005

Positive solutions to the equation x == 5 (mod 8). - K.V.Iyer, Apr 27 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 248.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016813, A017101, A017113, A110534, A113770.

Cf. A004776 (complement), A007521 (primes).

a004770 = (subtract 3) . (* 8)
a004770_list = [5, 13 ..]  -- Reinhard Zumkeller, Aug 17 2012

[8*n-3: n in [1..60]]; // Vincenzo Librandi, May 28 2011

Range[5, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
LinearRecurrence[{2,-1},{5,13},60] (* or *) NestList[#+8&,5,60] (* Harvey P. Dale, Jun 28 2021 *)

a(n)=8*n-3 \\ Charles R Greathouse IV, Sep 24 2015

[8*n-3 for n in range(1,57)] # Stefano Spezia, Jul 23 2025

From R. J. Mathar, Mar 14 2011: (Start)
a(n) = 8*n - 3.
G.f.: x*(5+3*x)/(x-1)^2. (End)
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(8*x - 3) + 3.
a(n) = A113770(n)/2 = A016813(2*n-1). (End)

A003629 Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p.

Original entry on oeis.org

3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557, 563

Offset: 1

N. J. A. Sloane, Mira Bernstein

Complement of A038873 relative to A000040.
Also primes p such that p divides 2^((p-1)/2) + 1. - Cino Hilliard, Sep 04 2004
Primes p such that p^2 == 25 (mod 48), n > 1. - Gary Detlefs, Dec 29 2011
This sequence gives the primes p which satisfy C(p, x = 0) = -1, where C(p, x) is the minimal polynomial of 2*cos(Pi/p) (see A187360). For a proof see a comment on C(n, 0) in A230075. - Wolfdieter Lang, Oct 24 2013
Except for the initial 3, these are the primes p such that Fibonacci(p) mod 6 = 5. - Gary Detlefs, May 26 2014
Inert rational primes in the field Q(sqrt(2)). - N. J. A. Sloane, Dec 26 2017
If a prime p is congruent to 3 or 5 (mod 8) and r > 1, then 2^((p-1)*p^(r-1)/2) == -1 (mod p^r). - Marina Ibrishimova, Sep 29 2018
For the proofs or the comments by Cino Hilliard and Marina Ibrishimova, see link below. - Robert Israel, Apr 24 2019

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Ronald S. Irving, Integers, Polynomials, and Rings. New York: Springer-Verlag (2004): 274.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Robert Israel, Proof of comments by Hilliard (also conjectured by Detlefs) and Ibrishimova
Index to sequences related to decomposition of primes in quadratic fields

Cf. A001132 (complement from the odd primes), A007521 (subsequence), A038873, A226523.

[3] cat [p: p in PrimesUpTo (600) | p^2 mod 48 eq 25]; // Vincenzo Librandi, May 23 2016

for n from 2 to 563 do if(ithprime(n)^2 mod 48 = 25) then print(ithprime(n)) fi od. # Gary Detlefs, Dec 29 2011

Select[Prime @ Range[2, 105], JacobiSymbol[2, # ] == -1 &] (* Robert G. Wilson v, Dec 15 2005 *)
Select[Union[8Range[100] - 5, 8Range[100] - 3], PrimeQ[#] &] (* Alonso del Arte, May 22 2016 *)
Select[Prime[Range[150]],MemberQ[{3,5},Mod[#,8]]&] (* Harvey P. Dale, Mar 02 2022 *)

is(n)=isprime(n) && (n%8==3 || n%8==5) \\ Charles R Greathouse IV, Mar 21 2016

A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

Original entry on oeis.org

5, 29, 1237, 32171803229, 829, 405565189, 14717, 39405395843265000967254638989319923697097319108505264560061, 282860648026692294583447078797184988636062145943222437, 53, 421, 13, 109, 4133, 6476791289161646286812333, 461, 34549, 453690033695798389561735541

Offset: 1

Labos Elemer, Oct 09 2000

nonn

			a(3) = 1237 = 8*154 + 5 is the smallest suitable prime divisor of (5*29)*5*29 + 4 = 21029 = 17*1237. (Although 17 is the smallest prime divisor, 17 is not congruent to 5 modulo 8.)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

P. G. L. Dirichlet, Supplement VI: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, 1871, 24 pages.

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A007521, A057204-A057208.

a={5}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4+q^2][[All,1]],Mod[#,8]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

lista(nn) = {v = vector(nn); v[1] = 5; print1(v[1], ", "); for (n=2, nn, f = factor(4 + prod(k=1, n-1, v[k])^2); for (k=1, #f~, if (f[k, 1] % 8 == 5, v[n] = f[k,1]; break);); print1(v[n], ", "););} \\ Michel Marcus, Oct 27 2014

More terms from Sean A. Irvine, Oct 26 2014

A105126 Primes of the form 16n+9.

Original entry on oeis.org

41, 73, 89, 137, 233, 281, 313, 409, 457, 521, 569, 601, 617, 761, 809, 857, 937, 953, 1033, 1049, 1097, 1129, 1193, 1289, 1321, 1433, 1481, 1609, 1657, 1721, 1753, 1801, 1913, 1993, 2089, 2137, 2153, 2281, 2297, 2377, 2393, 2441, 2473, 2521, 2617, 2633, 2713

Offset: 1

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

nonn

A prime of the form 16n+9 is represented by exactly one of x^2 + 32y^2 and x^2 + 64y^2 (see Kaplanski link). - Michel Marcus, Dec 23 2012

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

Cf. A002145, A007521, A105127-A105132.

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),t2]; fi; od; t1; end; f(4);

lst={};Do[p=16*n+9;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!,1}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)

select(n->n%16==9, primes(500)) \\ Charles R Greathouse IV, Apr 29 2015

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

A139487 Numbers k such that 8k + 7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 15, 18, 20, 23, 24, 27, 29, 32, 33, 38, 44, 45, 47, 53, 54, 57, 59, 60, 62, 74, 75, 78, 80, 89, 90, 92, 93, 102, 104, 107, 110, 113, 114, 120, 122, 123, 128, 129, 132, 135, 137, 143, 152, 153, 159, 162, 164, 165, 170, 174, 177, 179, 180, 183, 185

Offset: 1

Artur Jasinski, Apr 23 2008

For numbers k such that:
8k+1 is prime see A005123, primes see A007519;
8k+3 is prime see A005124, primes see A007520;
8k+5 is prime see A105133, primes see A007521;
8k+7 is prime see A139487, primes see A007522.
8k + 7 divides A000225(4k+3). - Jinyuan Wang, Mar 08 2019

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

Cf. A000225, A007519, A007520, A007521, A007522, A005123, A005124, A105133.

[n: n in [0..200] | IsPrime(8*n+7)]; // Vincenzo Librandi, Jun 25 2014

a = {}; Do[If[PrimeQ[8 n + 7], AppendTo[a, n]], {n, 0, 300}]; a
Select[Range[0,200],PrimeQ[8#+7]&] (* Harvey P. Dale, Oct 10 2012 *)

is(n)=isprime(8*n+7) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A007522(n) - 7)/8, n >= 1.

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A385224 Primes p such that multiplicative order of -4 modulo p is odd.

Original entry on oeis.org

5, 13, 29, 37, 41, 53, 61, 101, 109, 113, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 313, 317, 349, 373, 389, 397, 409, 421, 457, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 761, 773, 797, 809, 821, 829, 853, 857, 877, 941, 953, 997

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of -4 modulo a(n) is A385230(n).
Different from A133204: 593 is here but not in A133204, and 1601 is in A133204 but not here.
The sequence contains no primes congruent to 3 modulo 4 and all primes congruent to 5 modulo 8:
  - If p is a term of this sequence, then -4 is a quadratic residue modulo p, so p == 1 (mod 4);
  - For p == 1 (mod 4), we have (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i is a solution to i^2 == -1 (mod p).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A002144 (primes congruent to 1 modulo 4).
Contains A007521 (primes congruent to 5 or modulo 8) as a proper subsequence.
Cf. A385230 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), this sequence (base -4), A385225 (base -5).
Cf. A133204.

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A004770 Numbers of the form 8k+5; or, numbers whose binary expansion ends in 101.

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A003629 Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p.

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A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

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Mathematica

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A105126 Primes of the form 16n+9.

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Maple

Mathematica

PARI

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

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Magma

Maple

Mathematica

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