A002143 -id:A002143 - 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

A076409 Sum of the quadratic residues of prime(n).

Original entry on oeis.org

1, 1, 5, 7, 22, 39, 68, 76, 92, 203, 186, 333, 410, 430, 423, 689, 767, 915, 1072, 994, 1314, 1343, 1577, 1958, 2328, 2525, 2369, 2675, 2943, 3164, 3683, 3930, 4658, 4587, 5513, 5134, 6123, 6520, 6012, 7439, 7518, 8145, 7831, 9264, 9653, 8955, 10761, 11596

Offset: 1

R. K. Guy, Oct 08 2002

Row sums of A063987. - R. J. Mathar, Jan 08 2015

prime(n) divides a(n) for n > 2. This is implied by a variant of Wolstenholme's theorem (see Hardy & Wright reference). - Isaac Saffold, Jun 21 2018

			If n = 3, then p = 5 and a(3) = 1 + 4 = 5. If n = 4, then p = 7 and a(4) = 1 + 4 + 2 = 7. If n = 5, then p = 11 and a(5) = 1 + 4 + 9 + 5 + 3 = 22. - _Michael Somos_, Jul 01 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 4th ed., Oxford Univ. Press, 1960, p. 88-90.
Kenneth A. Ribet, Modular forms and Diophantine questions, Challenges for the 21st century (Singapore 2000), 162-182; World Sci. Publishing, River Edge NJ 2001; Math. Rev. 2002i:11030.

Zak Seidov, Table of n, a(n) for n = 1..1000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Cf. A076410.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

A076409 := proc(n)
  local a,p,i ;
  p := ithprime(n) ;
  a := 0 ;
  for i from 1 to p-1 do
    if numtheory[legendre](i,p) = 1 then
       a := a+i ;
    end if;
  end do;
  a ;
end proc: # R. J. Mathar, Feb 26 2011

Join[{1,1}, Table[ Apply[ Plus, Flatten[ Position[ Table[ JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], {n, 3, 48}]]
Join[{1}, Table[p=Prime[n]; If[Mod[p,4]==1, p(p-1)/4, Sum[PowerMod[k,2, p],{k,p/2}]], {n,2,1000}]] (* Zak Seidov, Nov 02 2011 *)
a[ n_] := If[ n < 3, Boole[n > 0], With[{p = Prime[n]}, Sum[ Mod[k^2, p], {k, (p - 1)/2}]]]; (* Michael Somos, Jul 01 2018 *)

a(n,p=prime(n))=if(p<5,return(1)); if(k%4==1, return(p\4*p)); sum(k=1,p-1,k^2%p) \\ Charles R Greathouse IV, Feb 21 2017

If prime(n) = 4k+1 then a(n) = k*(4k+1).

For n>2 if prime(n) = 4k+3 then a(n) = (k - b)*(4k+3) where b = (h(-p) - 1) / 2; h(-p) = A002143. For instance. If n=5, p=11, k=2, b=(1-1)/2=0 and a(5) = 2*11 = 22. If n=20, p=71, k=17, b=(7-1)/2=3 and a(20) = 14*71 = 994. - Andrés Ventas, Mar 01 2021

Edited and extended by Robert G. Wilson v, Oct 09 2002

A002148 Smallest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

3, 59, 131, 251, 419, 659, 1019, 971, 1091, 2099, 1931, 1811, 3851, 3299, 2939, 3251, 4091, 4259, 8147, 5099, 9467, 6299, 6971, 8291, 8819, 14771, 22619, 9539, 13331, 18443, 11171, 16979, 12011, 13859, 16931, 17939, 28211, 19211, 24251, 20411

Offset: 0

N. J. A. Sloane and Mira Bernstein

nonn

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
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 Broadhurst and T. D. Noe, Table of n, a(n) for n = 0..10399
D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

Cf. A002143 (class numbers), A002149, A003173, A006203.

a=Table[0, {101}]; Do[If[PrimeQ[m], c=NumberFieldClassNumber[Sqrt[-m]]; If[c<102 && a[[c]]==0, a[[c]]=m]], {m, 3, 30000, 8}]; Table[a[[n]], {n, 1, 101, 2}]

More terms from Robert G. Wilson v, Apr 17 2001

Edited by Dean Hickerson, Mar 17 2003

A002146 Smallest prime == 7 (mod 8) where Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

7, 23, 47, 71, 199, 167, 191, 239, 383, 311, 431, 647, 479, 983, 887, 719, 839, 1031, 1487, 1439, 1151, 1847, 1319, 3023, 1511, 1559, 2711, 4463, 2591, 2399, 3863, 2351, 3527, 3719

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Conjecture: a(n) < A002148(n) for all n >= 1. - Jianing Song, Jul 20 2022

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
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 Broadhurst and T. D. Noe, Table of n, a(n) for n = 0..28603
D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

Cf. A002147, A002148, A060651, A002143 (class numbers).

a(n) = forprime(p=2, oo, if ((p % 8) == 7, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022

A165951 a(n) = (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 3, 0, 3, 0, 0, 1, 5, 0, 3, 0, 1, 7, 0, 5, 3, 0, 0, 0, 5, 3, 0, 0, 5, 5, 0, 3, 0, 7, 0, 1, 11, 0, 5, 0, 13, 0, 0, 9, 3, 7, 5, 0, 0, 15, 0, 7, 0, 13, 0, 11, 0, 0, 3, 0, 3, 19, 0, 0, 3, 0, 5, 0, 0, 19, 9, 0, 3, 17, 0, 0, 0, 0, 9, 0, 21, 0, 15, 5, 0, 0, 0, 7, 7, 25, 7, 9, 3, 21, 0, 0

Offset: 1

Christopher Hunt Gribble, Oct 01 2009

nonn

The positive terms are A002143 minus its first term (for p=3, A002143(1)=1 corresponds to 0 here). - Javier Múgica, Nov 23 2024

C. H. Gribble, Table of n, a(n) for n=1,..., 348513 (i.e. for primes < 5x10^6).

Cf. A076410, A125616.

Cf. allso A002143 (positive terms).

a(n) = Sum_{j=1..prime(n)-1} floor(j^2/prime(n)) - floor((prime(n)-2)*(prime(n)-1)/3) for n >= 3.

a(n) = A125616(n) - A076410(n) for n>=3.

a(1)-a(2) added by Christopher Hunt Gribble, Oct 07 2009

A228131 a(n) = Sum_{i=0..n-1} K(i,n)*i, where K(,) is Kronecker symbol.

Original entry on oeis.org

0, 1, -1, 4, 0, 6, -7, 0, 27, 6, -11, -8, 0, 20, -30, 64, 0, -4, -19, 0, 0, 46, -69, -48, 250, 106, -9, 0, 0, 68, -93, 0, 0, 44, -70, 216, 0, 82, -156, 0, 0, 60, -43, -88, 0, 148, -235, -32, 1029, 94, -102, 0, 0, 6, -220, -224, 0, -82, -177, 0, 0, 168, -126, 1024, 0, 304, -67, 0, 0, 268, -497, 0, 0, 494, -50, -152, 0, 276, -395, 0, 2187, 4, -249, 0, 0, 310, -522, -176, 0, 388, -182, 0, 0, 424, -760, -192, 0, 202, 0, 2000

Offset: 1

Max Alekseyev, Aug 11 2013

sign

For prime n==1 (mod 4), a(n) = 0.

For prime n==3 (mod 4) and n>3, i.e., n=A002145(m) for m>1, a(n) = -n*A002143(m).

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

Cf. A255643, A255644

Table[Sum[KroneckerSymbol[k, n]*k, {k, 0, n - 1}], {n, 0, 50}] (* G. C. Greubel, Apr 23 2018 *)

PARI
```
a(n) = sum(i=1,n-1, kronecker(i,n)*i)
```

A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n).

Original entry on oeis.org

1, 0, 1, 3, 0, 0, 3, 3, 0, 3, 0, 0, 3, 5, 0, 9, 0, 3, 7, 0, 5, 9, 0, 0, 0, 5, 9, 0, 0, 5, 15, 0, 9, 0, 7, 0, 3, 11, 0, 15, 0, 13, 0, 0, 9, 9, 7, 15, 0, 0, 15, 0, 21, 0, 13, 0, 11, 0, 0, 9, 0, 9, 19, 0, 0, 9, 0, 15, 0, 0, 19, 9, 0, 9, 17, 0, 0, 0, 0, 27, 0, 21, 0, 15, 15, 0, 0, 0, 7, 21, 25, 7, 27, 9, 21, 0

Offset: 2

T. D. Noe, May 21 2010

When prime(n)=1 (mod 4), then a(n)=0. When prime(n)=3 (mod 4), then a(n)>0. When prime(n)=3 (mod 8) and prime(n)>3, then 3 divides a(n). See Borevich and Shafarevich. The nonzero terms of this sequence are closely related to A002143, the class number of primes p=3 (mod 4).

Same as difference between the numbers of quadratic residues and nonresidues (mod p) less than p/2, where p=prime(n). - Jonathan Sondow, Oct 30 2011

			The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Hence a(8)=6-3=3.

Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966, p. 346.
H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51.

R. J. Mathar, Table of n, a(n) for n = 2..2066
MathOverflow, Most squares in the first half-interval

Cf. A002143, A178154 (without the zero terms).

A178153 := proc(n)
    local r,a,p;
    p := ithprime(n) ;
    a := 0 ;
    for r from 1 to p/2 do
        if numtheory[legendre](r,p) =1 then
            a := a+1 ;
        end if;
    end do:
    for r from ceil(p/2) to p-1 do
        if numtheory[legendre](r,p) =1 then
            a := a-1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Feb 10 2017

Table[p=Prime[n]; Length[Select[Range[(p-1)/2], JacobiSymbol[ #,p]==1&]] - Length[Select[Range[(p+1)/2,p-1], JacobiSymbol[ #,p]==1&]], {n,2,100}]
Table[p = Prime[n]; Sum[ JacobiSymbol[a, p], {a, 1, (p-1)/2}], {n, 2, 100}] (* Jonathan Sondow, Oct 30 2011 *)

a(n) = A178151(n) - A178152(n).

a(n) = Sum_{j=1..(p-1)/2} (j|p), where p = prime(n) and (j|p) = +-1 is the Legendre symbol. - Jonathan Sondow, Oct 30 2011

cp's OEIS Frontend

A111945 Duplicate of A002143.

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A002145 Primes of the form 4*k + 3.

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A076409 Sum of the quadratic residues of prime(n).

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A002148 Smallest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.

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A002146 Smallest prime == 7 (mod 8) where Q(sqrt(-p)) has class number 2n+1.

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A165951 a(n) = (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic residues of prime(n)) / prime(n).

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A228131 a(n) = Sum_{i=0..n-1} K(i,n)*i, where K(,) is Kronecker symbol.

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A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n).

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