A095278 -id:A095278 - cp's OEIS frontend

A002145 Primes of the form 4*k + 3.

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571

Offset: 1

N. J. A. Sloane

Or, odd primes p such that -1 is not a square mod p, i.e., the Legendre symbol (-1/p) = -1. [LeVeque I, p. 66]. - N. J. A. Sloane, Jun 28 2008
Primes which are not the sum of two squares, see the comment in A022544. - Artur Jasinski, Nov 15 2006
Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)
Inert rational primes in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1. - Benoit Cloitre, Oct 22 2002
For p and q both belonging to the sequence, exactly one of the congruences x^2 = p (mod q), x^2 = q (mod p) is solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Also primes p that divide L((p-1)/2) or L((p+1)/2), where L(n) = A000032(n), the Lucas numbers. Union of A122869 and A122870. - Alexander Adamchuk, Sep 16 2006
Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - Alexander Adamchuk, Nov 30 2006
Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - Alexander Adamchuk, Apr 18 2007
This sequence is a proper subset of the set of the absolute values of negative fundamental discriminants (A003657). - Paul Muljadi, Mar 29 2008
Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after Paul Curtz, Sep 10 2008
A079261(a(n)) = 1; complement of A145395. - Reinhard Zumkeller, Oct 12 2008
Subsequence of A007970. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = -1.
Primes p such that p XOR 2 = p - 2. Brad Clardy, Oct 25 2011 (Misleading in the sense that this is a formula for the super-sequence A004767. - R. J. Mathar, Jul 28 2014)
It appears that each term of A004767 is the mean of two terms of this subsequence of primes therein; cf. A245203. - M. F. Hasler, Jul 13 2014
Numbers n > 2 such that ((n-2)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 24 2016
Odd numbers n > 1 such that ((n-1)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 25 2016
Primes p such that (p-2)!! == (p-3)!! (mod p). - Thomas Ordowski, Jul 28 2016
See Granville and Martin for a discussion of the relative numbers of primes of the form 4k+1 and 4k+3. - Editors, May 01 2017
Sometimes referred to as Blum primes for their connection to A016105 and the Blum Blum Shub generator. - Charles R Greathouse IV, Jun 14 2018
Conjecture: a(n) for n > 4 can be written as a sum of 3 primes of the form 4k+1, which would imply that primes of the form 4k+3 >= 23 can be decomposed into a sum of 6 nonzero squares. - Thomas Scheuerle, Feb 09 2023

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 146-147.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 252.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 66.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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, NY, 1986, Revised edition 1987. See p. 90.

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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].
Dario Alpern, Gaussian primes.
Lenore Blum, Manuel Blum, and Mike Shub, A simple unpredictable pseudo-random number generator, SIAM Journal on Computing 15:2 (1 May 1986), pp. 364-383.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 67.
Harry J. Smith, Gaussian Primes.
Ian Stewart, The Great Mathematical Problems, 2013.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wolfram Research, The Gauss Reciprocity Law.
Index entries for Gaussian integers and primes.
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A000032, A002144, A003657, A085992, A122869, A122870, A334912.
Cf. A000408, A005098, A095278, A016754.
Apart from initial term, same as A045326.
Cf. A016105.
Cf. A004614 (multiplicative closure).

a002145 n = a002145_list !! (n-1)
a002145_list = filter ((== 1) . a010051) [3, 7 ..]
-- Reinhard Zumkeller, Aug 02 2015, Sep 23 2011

[4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // Arkadiusz Wesolowski, Nov 15 2013

A002145 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        a := nextprime(procname(n-1)) ;
        while a mod 4 <>  3 do
            a := nextprime(a) ;
        end do;
        return a;
    end if;
end proc:
seq(A002145(n),n=1..20) ; # R. J. Mathar, Dec 08 2011

Select[4Range[150] - 1, PrimeQ] (* Alonso del Arte, Dec 19 2013 *)
Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] == 1 &] (* or *)
Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == -1 &] (* Robert G. Wilson v, May 11 2014 *)

forprime(p=2,1e3,if(p%4==3,print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

def A002145_list(n): return [p for p in prime_range(1, n + 1) if p % 4 == 3]  # Peter Luschny, Jul 29 2014

Remove from A000040 terms that are in A002313.
Intersection of A000040 and A004767. - Alonso del Arte, Apr 22 2014
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A243379.
Product_{k>=1} (1 + 1/a(k)^2) = A243381.
Product_{k>=1} (1 - 1/a(k)^3) = A334427.
Product_{k>=1} (1 + 1/a(k)^3) = A334426.
Product_{k>=1} (1 - 1/a(k)^4) = A334448.
Product_{k>=1} (1 + 1/a(k)^4) = A334447.
Product_{k>=1} (1 - 1/a(k)^5) = A334452.
Product_{k>=1} (1 + 1/a(k)^5) = A334451. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/a(k)) / (1 + 1/A002144(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/a(k)) / (1 - 1/A002144(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log(2 * (2^(n*s) - 1) * (n*s - 1)! * zeta(n*s) / (Pi^(n*s) * abs(EulerE(n*s - 1))))/n, s >= 3 odd number. - Dimitris Valianatos, May 20 2020

More terms from James Sellers, Apr 21 2000

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

A095277 Numbers k such that 4k + 3 is composite.

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 15, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 35, 36, 38, 39, 42, 43, 45, 46, 48, 50, 51, 53, 54, 57, 58, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85, 87, 88, 90, 92, 93, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 108

Offset: 1

Antti Karttunen, Jun 01 2004

Terms can be written as (4xy +- (x-y)) - 1 for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

Numbers k such that (4*k)!/(4*k + 3) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the positive terms in the following triangular array:
  *;
  3,  *;
  *,  8,  *;
  6,  *, 15,  *;
  *, 13,  *, 24,  *;
  9,  *, 22,  *, 35,  *;
  *, 18,  *, 33,  *, 48,  *;
etc., where * marks the noninteger values of (2*h*k + k + h-1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Apr 22 2014

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

Complement of A095278. Cf. also A045751, A014076, A153170, A153088, A153329, A153343.

[n: n in [0..110] |not IsPrime(4*n+3)]; // Vincenzo Librandi, Apr 22 2014\

for n from 0 to 100 do
if irem(factorial(4*n), 4*n+3) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[150],!PrimeQ[4#+3]&] (* Harvey P. Dale, Jul 04 2011 *)

is(n)=!isprime(4*n+3) \\ Charles R Greathouse IV, Aug 01 2016

a(n) = (A091236(n) - 3)/4.

A123986 Numbers n for which 4n+1 and 4n+3 are primes.

Original entry on oeis.org

1, 4, 7, 10, 25, 34, 37, 49, 67, 70, 115, 130, 142, 154, 160, 202, 205, 214, 220, 262, 265, 307, 319, 322, 325, 370, 424, 430, 469, 487, 499, 520, 532, 535, 559, 577, 595, 637, 664, 682, 697, 700, 742, 814, 832, 847, 865, 889, 895, 955, 979, 982, 1000, 1012, 1039

Offset: 1

Artur Jasinski, Oct 30 2006

nonn

All terms == 1 mod 3. - Zak Seidov, Dec 02 2011

Intersection of A005098 and A095278. - Michel Marcus, Jan 31 2015

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

Cf. A005098, A005099, A095278.

[n: n in [0..1100] |IsPrime(4*n+1) and IsPrime(4*n+3)]; // Vincenzo Librandi, Feb 01 2015

Select[Range[1100], And @@ PrimeQ /@ ({1, 3} + 4#) &] (* Ray Chandler, Nov 05 2006 *)
nn=10000;k=0;x=1;re=Reap[While[kZak Seidov, Dec 02 2011 *)

Extended by Ray Chandler, Nov 05 2006

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

A111199 Numbers k such that 4k + 9 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 20, 22, 23, 25, 26, 32, 35, 37, 41, 43, 46, 47, 55, 56, 58, 62, 65, 67, 68, 71, 76, 77, 82, 85, 86, 91, 95, 97, 98, 100, 103, 106, 110, 112, 113, 125, 128, 133, 137, 140, 142, 146, 148, 151, 152, 158, 161, 163, 166, 167, 173, 175, 181, 187

Offset: 1

Parthasarathy Nambi, Oct 24 2005

nonn

			For k=98, 4*k + 9 = 401 (prime).

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

Cf. A005098, A095278, A089986, A153238, A144562.

is(n)=isprime(4*n+9) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005098(n+1) - 2. - R. J. Mathar, Sep 23 2009

More terms from R. J. Mathar, Sep 23 2009

A089986 Numbers n such that 4n + 7 is prime.

Original entry on oeis.org

-1, 0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 19, 24, 25, 30, 31, 33, 36, 39, 40, 43, 46, 48, 51, 54, 55, 58, 61, 64, 66, 69, 75, 76, 81, 85, 88, 90, 93, 94, 103, 106, 108, 109, 114, 115, 118, 120, 121, 123, 124, 129, 135, 139, 141, 145, 148, 150, 153, 156, 159, 160, 163, 169

Offset: 1

Giovanni Teofilatto, Jan 13 2004

sign

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

Cf. A005099 ((( Primes = -1 mod 4 ) + 1)/4), A005098 (4n+1 is prime), A095278 (4n+3 is prime), A111215 (4n+5 is prime).

[ n: n in [-1..170] | IsPrime(4*n+7) ];

A089986:=n->`if`(isprime(4*n+7), n, NULL): seq(A089986(n), n=-1..200); # Wesley Ivan Hurt, Sep 18 2014

Select[Range[-1, 200], PrimeQ[4 # + 7] &] (* Wesley Ivan Hurt, Sep 18 2014 *)

is(n)=isprime(4*n+7) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005099(n) - 2 = A095278(n) - 1.

Edited and extended by Klaus Brockhaus, Dec 22 2008

cp's OEIS Frontend

A002145 Primes of the form 4*k + 3.

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

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A095277 Numbers k such that 4k + 3 is composite.

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A123986 Numbers n for which 4n+1 and 4n+3 are primes.

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

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A111199 Numbers k such that 4k + 9 is prime.

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A089986 Numbers n such that 4n + 7 is prime.

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