A334448 -id:A334448 - 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

A334427 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^3).

Original entry on oeis.org

9, 5, 9, 1, 4, 2, 7, 1, 1, 0, 4, 3, 2, 0, 7, 3, 4, 4, 9, 9, 9, 7, 0, 5, 9, 1, 3, 7, 5, 0, 2, 0, 9, 8, 1, 5, 3, 6, 5, 4, 2, 3, 6, 5, 9, 7, 7, 4, 4, 5, 7, 1, 0, 6, 3, 4, 8, 6, 2, 6, 6, 4, 3, 2, 8, 0, 6, 8, 5, 4, 9, 8, 8, 3, 8, 6, 4, 2, 2, 3, 8, 9, 3, 4, 1, 2, 3, 9, 3, 7, 7, 5, 3, 7, 4, 3, 9, 7, 1, 3, 5, 8, 1, 1, 1, 3

Offset: 0

Vaclav Kotesovec, Apr 30 2020

			0.959142711043207344999705913750209815365423...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 3 3 = 1/A334427).

Cf. A002145, A243379, A334448, A334452.

A334426 / A334427 = 28*zeta(3)/Pi^3.

A334425 * A334427 = 8/(7*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

A334446 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

Original entry on oeis.org

9, 9, 8, 3, 5, 0, 4, 9, 5, 7, 2, 3, 2, 0, 0, 4, 0, 6, 4, 9, 9, 9, 0, 5, 5, 1, 7, 5, 6, 5, 5, 4, 1, 6, 2, 9, 1, 9, 1, 5, 3, 9, 4, 0, 7, 0, 1, 9, 6, 0, 5, 7, 9, 5, 0, 4, 6, 3, 1, 4, 1, 5, 8, 5, 0, 4, 2, 4, 1, 6, 7, 8, 3, 5, 9, 9, 8, 8, 2, 2, 5, 7, 2, 3, 4, 0, 8, 8, 7, 8, 4, 3, 7, 0, 3, 6, 8, 2, 4, 7, 8, 8, 1, 1, 3, 7

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

For s>1, zeta(s, 1/4) - zeta(s, 3/4) = (-1)^s*(PolyGamma(s-1, 1/4) - PolyGamma(s-1, 3/4))/(s-1)! = 2*(-1)^s * PolyGamma(s-1, 1/4) / Gamma(s) - 2^s*(2^s - 1)*zeta(s) = 4^s * DirichletBeta(s).

			0.998350495723200406499905517565541629191539407019605795046314...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 1 4 = 1/A334446).

Cf. A002144, A088539, A334425, A334450.

A334445 / A334446 = 35*(PolyGamma(3, 1/4)/(8*Pi^4) - 1)/34.

A334446 * A334448 = 96/Pi^4.

More digits from Vaclav Kotesovec, Jun 27 2020

A155918 Number of squared hypotenuses mod n in two dimensions.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 23, 15, 25, 26, 21, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 27, 43, 50, 51, 39, 53, 42, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 71, 35, 73

Offset: 1

Steven Finch, Jan 30 2009

Number of images of the map (x,y) -> x^2+y^2 in Z_n.

Let n = p^e and k = r*p^b (gcd(r,p) = 1). If p == 1 (mod 4), then x^2 + y^2 == k (mod p) always have solutions; if p == 3 (mod 4), then x^2 + y^2 == k (mod p) is solvable if and only if b is even or b >= e; if p = 2, then x^2 + y^2 == k (mod p) is solvable if and only if r == 1 (mod 4) or b >= e - 1. If 0 <= k < n, then the number of solutions to x^2 + y^2 == k (mod n) is A305191(n,k). - Jianing Song, Apr 20 2019

Jianing Song, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Cf. A305191, A334427, A334448.

(For[v = Table[0, {m, 1, n^2}]; m = 1; i = 0, i < n, i++, For[j = 0, j < n, j++, v[[m]] = Mod[i^2 + j^2, n]; m = m + 1]]; Length[Union[v]])
(* Second program: *)
a[n_] := Module[{p, e}, Product[{p, e} = pe; Which[Mod[p, 4] == 1, p^e, Mod[p, 4] == 3, Ceiling[p^(e+1)/(p+1)], p == 2, 2^(e-1) + 1, True, p], {pe, FactorInteger[n]}]];
Array[a, 100] (* Jean-François Alcover, Jul 30 2020 *)

a(n) = #Set(vector(n^2, i, ((i%n)^2 + (i\n)^2) % n)); \\ Michel Marcus, Jul 08 2017

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==2, r*=2^(e-1)+1);
        if(p%4==1, r*=p^e);
        if(p%4==3, r*=ceil(p^(e+1)/(p+1)));
    );
    return(r);
} \\ Jianing Song, Apr 20 2019

Multiplicative with a(p^e) = p^e if p == 1 (mod 4); ceiling(p^(e+1)/(p+1)) if p == 3 (mod 4); 2^(e-1) + 1 if p = 2. - Jianing Song, Apr 20 2019

Sum_{k=1..n} a(k) ~ c * n^2, where c = (11/24) * Product_{p prime == 3 (mod 4)} (1 - 1/p^3)/(1 - 1/p^4) = (11/24) * A334427/A334448 = 0.44532386516028771931... . - Amiram Eldar, Feb 17 2024

A334447 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

Original entry on oeis.org

1, 0, 1, 2, 8, 4, 9, 7, 3, 7, 5, 0, 3, 6, 5, 8, 2, 4, 1, 0, 5, 3, 7, 3, 8, 8, 0, 9, 6, 3, 0, 1, 1, 2, 0, 3, 9, 6, 8, 4, 5, 0, 4, 2, 1, 6, 5, 5, 3, 8, 6, 9, 4, 5, 0, 9, 2, 2, 2, 1, 4, 4, 1, 8, 1, 9, 1, 3, 4, 1, 5, 6, 6, 9, 0, 0, 5, 5, 2, 5, 7, 1, 6, 6, 4, 2, 4, 8, 6, 1, 2, 7, 5, 4, 1, 3, 0, 2, 9, 9, 9, 3, 4, 4, 9

Offset: 1

Vaclav Kotesovec, Apr 30 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)) = 1 / (2 * (-1)^s * PolyGamma(s-1, 1/4) / (2^s * (2^s - 1) * Gamma(s) * zeta(s)) - 1).

			1.01284973750365824105373880963011203968450421655386945092221...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.

Cf. A002145, A243381, A334426, A334451.

A334447 / A334448 = 1/(PolyGamma(3, 1/4)/(8*Pi^4) - 1).

A334445 * A334447 = 1680 / (17*Pi^4).

More digits from Vaclav Kotesovec, Jun 27 2020

A334452 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^5).

Original entry on oeis.org

9, 9, 5, 8, 1, 8, 7, 2, 9, 8, 6, 8, 0, 8, 0, 5, 9, 5, 9, 4, 3, 3, 8, 5, 1, 6, 1, 6, 4, 3, 1, 6, 5, 9, 7, 1, 8, 7, 4, 3, 4, 7, 2, 7, 3, 1, 8, 4, 9, 1, 0, 5, 6, 6, 3, 9, 8, 3, 5, 7, 7, 1, 4, 6, 9, 8, 0, 3, 9, 6, 3, 9, 6, 7, 0, 3, 1, 0, 4, 6, 7, 9, 7, 0, 0, 5, 4, 4, 0, 1, 9, 6, 8, 0, 3, 1, 8, 2, 3, 3, 9, 3, 9, 8, 4, 5

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002145(k)^(2*s+1))/(1 - 1/A002145(k)^(2*s+1)) = (2*s)! * (2^(2*s + 2) - 2) * zeta(2*s+1) / (Pi^(2*s+1) * A000364(s)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)).

			0.99581872986808059594338516164316597187434727318491056639835771469803963967031...

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 4 3 5 = 1/A334452).

Cf. A002145, A243379, A334427, A334448.

A334451 / A334452 = 1488*zeta(5)/(5*Pi^5).

A334450 * A334452 = 32/(31*zeta(5)).

More digits from Vaclav Kotesovec, Jun 27 2020

A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

Original entry on oeis.org

1, 4, 2, 6, 5, 5, 6, 0, 6, 3, 5, 1, 2, 5, 9, 2, 8, 7, 8, 6, 9, 6, 8, 0, 9, 3, 1, 6, 1, 5, 5, 0, 8, 1, 6, 3, 6, 1, 2, 7, 6, 6, 9, 3, 6, 3, 6, 7, 7, 0, 3, 9, 0, 2, 8, 8, 7, 9, 9, 2, 2, 3, 0, 4, 4, 1, 2, 9, 6, 0, 4, 5, 2, 8, 6, 1, 5, 1, 9, 0, 1, 9, 1, 4, 6, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an equivalent of zeta(2) (= Pi^2/6 = A013661), where Euler's zeta(2) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

Close to the value of e^(3/2)/Pi.

			1.4265560635125928786968093161550816361276693636770...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025, A175647, A243379, A334448.

Cf. A344124, A344125.

Equals Sum_{i > 0} 1/A001481(i)^2.
Equals Product_{i > 0} 1/(1-A055025(i)^-2).
Equals 1/(1-prime(1)^(-2)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-2)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-4)), where prime(n) = A000040(n).
Equals (4/3)/(A243379*A334448).
Equals zeta_{2,0} (2) * zeta_{4,1} (2) * zeta_{4,3} (4), where zeta_{4,1} (2)  = A175647 and zeta_{2,0} (s) = 2^s/(2^s - 1).

cp's OEIS Frontend

A002145 Primes of the form 4*k + 3.

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A334427 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^3).

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A334446 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

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A155918 Number of squared hypotenuses mod n in two dimensions.

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A334447 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

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A334452 Decimal expansion of Product_{k>=1} (1 - 1/A002145(k)^5).

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A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

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