A088539 -id:A088539 - cp's OEIS frontend

A002144 Pythagorean primes: primes of the form 4*k + 1.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617

Offset: 1

N. J. A. Sloane

Rational primes that decompose in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
These are the prime terms of A009003.
-1 is a quadratic residue mod a prime p if and only if p is in this sequence.
Sin(a(n)*Pi/2) = 1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002
If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2 = p (mod q), x^2 = q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Odd primes such that binomial(p-1, (p-1)/2) == 1 (mod p). - Benoit Cloitre, Feb 07 2004
Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n), A002366(n), a(n)}.
Also, primes of the form a^k + b^k, k > 1. - Amarnath Murthy, Nov 17 2003
The square of a(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = a(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005
Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk, Aug 10 2006
The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi, Mar 28 2008
A079260(a(n)) = 1; complement of A137409. - Reinhard Zumkeller, Oct 11 2008
From Artur Jasinski, Dec 10 2008: (Start)
If we take 4 numbers: 1, A002314(n), A152676(n), A152680(n) then multiplication table modulo a(n) is isomorphic to the Latin square:
   1 2 3 4
   2 4 1 3
   3 1 4 2
   4 3 2 1
and isomorphic to the multiplication table of {1, i, -i, -1} where i is sqrt(-1), A152680(n) is isomorphic to -1, A002314(n) with i or -i and A152676(n) vice versa -i or i. 1, A002314(n), A152676(n), A152680(n) are subfield of Galois field [a(n)]. (End)
Primes p such that the arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes: this one and A002145. - Ctibor O. Zizka, Oct 20 2009
Equivalently, the primes p for which the smallest extension of F_p containing the square roots of unity (necessarily F_p) contains the 4th roots of unity. In this respect, the n = 2 case of a family of sequences: see n=3 (A129805) and n=5 (A172469). - Katherine E. Stange, Feb 03 2010
Subsequence of A007969. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = 1.
k^k - 1 is divisible by 4*k + 1 if 4*k + 1 is a prime (see Dickson reference). - Gary Detlefs, May 22 2013
Not only are the squares of these primes the sum of two nonzero squares, but the primes themselves are also. 2 is the only prime equal to the sum of two nonzero squares and whose square is not. 2 is therefore not a Pythagorean prime. - Jean-Christophe Hervé, Nov 10 2013
The statement that these primes are the sum of two nonzero squares follows from Fermat's theorem on the sum of two squares. - Jerzy R Borysowicz, Jan 02 2019
The decompositions of the prime and its square into two nonzero squares are unique. - Jean-Christophe Hervé, Nov 11 2013. See the Dickson reference, Vol. II, (B) on p. 227. - Wolfdieter Lang, Jan 13 2015
p^e for p prime of the form 4*k+1 and e >= 1 is the sum of 2 nonzero squares. - Jon Perry, Nov 23 2014
Primes p such that the area of the isosceles triangle of sides (p, p, q) for some integer q is an integer. - Michel Lagneau, Dec 31 2014
This is the set of all primes that are the average of two squares. - Richard R. Forberg, Mar 01 2015
Numbers k such that ((k-3)!!)^2 == -1 (mod k). - Thomas Ordowski, Jul 28 2016
This is a subsequence of primes of A004431 and also of A016813. - Bernard Schott, Apr 30 2022
In addition to the comment from Jean-Christophe Hervé, Nov 10 2013: All powers as well as the products of any of these primes are the sum of two nonzero squares. They are terms of A001481, which is closed under multiplication. - Klaus Purath, Nov 19 2023

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2 + d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2 - a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
  ---------------------------------
   p  a  b  t_1  c   d t_2 t_3  t_4
  ---------------------------------
   5  1  2   1   3   4   4   3    6
  13  2  3   3   5  12  12   5   30
  17  1  4   2   8  15   8  15   60
  29  2  5   5  20  21  20  21  210
  37  1  6   3  12  35  12  35  210
  41  4  5  10   9  40  40   9  180
  53  2  7   7  28  45  28  45  630
  ...
a(7) = 53 = A002972(7)^2 + (2*A002973(7))^2 = 7^2 + (2*1)^2 = 49 + 4, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. E. Dickson, "History of the Theory of Numbers", Chelsea Publishing Company, 1919, Vol I, page 386
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 241, 243.
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.
C. Banderier, Calcul de (-1/p).
J. Butcher, Mathematical Miniature 8: The Quadratic Residue Theorem, NZMS Newsletter, No. 75, April 1999.
Hing Lun Chan, Windmills of the minds: an algorithm for Fermat's Two Squares Theorem, arXiv:2112.02556 [cs.LO], 2021.
R. Chapman, Quadratic reciprocity.
A. David Christopher, A partition-theoretic proof of Fermat's Two Squares Theorem, Discrete Mathematics, Volume 339, Issue 4, 6 April 2016, Pages 1410-1411.
J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem, The American Mathematical Monthly, Vol. 90, No. 9 (Nov., 1983), pp. 635-637.
Bernard Frénicle de Bessy, Traité des triangles rectangles en nombres : dans lequel plusieurs belles propriétés de ces triangles sont démontrées par de nouveaux principes, Michalet, Paris (1676) pp. 0-116; see p. 44, Consequence II.
Bernard Frénicle de Bessy, Méthode pour trouver la solution des problèmes par les exclusions. Abrégé des combinaisons. Des Quarrez magiques, in "Divers ouvrages de mathématiques et de physique, par MM. de l'Académie royale des sciences", (1693) "Troisième exemple", pp. 17-26, see in particular p. 25.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
D. & C. Hazzlewood, Quadratic Reciprocity.
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.
R. C. Laubenbacher and D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem, In: Anderson M, Katz V, Wilson R, eds. Who Gave You the Epsilon?: And Other Tales of Mathematical History. Spectrum. Mathematical Association of America; 2009:309-312.
R. C. Laubenbacher and D. J. Pengelley, Gauss, Eisenstein and the 'third' proof of the Quadratic Reciprocity Theorem, The Mathematical Intelligencer 16, 67-72 (1994).
K. Matthews, Serret's algorithm based Server.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
Carlos Rivera, Puzzle 968. Another property of primes 4m+1, The Prime Puzzles & Problems Connection.
D. Shanks, Review of "K. E. Kloss et al., Class number of primes of the form 4n+1", Math. Comp., 23 (1969), 213-214. [Annotated scanned preprint of review]
S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
Eric Weisstein's World of Mathematics, Wilson's Theorem.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Wikipedia, Quadratic reciprocity
Wolfram Research, The Gauss Reciprocity Law.
G. Xiao, Two squares.
D. Zagier, A One-Sentence Proof That Every Prime p == 1 (mod 4) Is a Sum of Two Squares, Am. Math. Monthly, Vol. 97, No. 2 (Feb 1990), p. 144. [From _Wolfdieter Lang_, Jan 17 2015 (thanks to Charles Nash)]
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A002145, A002314, A002476, A002972, A002973, A003658, A004431, A007519, A010051, A016813, A076339, A094407.
Cf. A114200, A133870, A142925, A152676, A152680, A173330, A173331, A208177, A208178, A334912.
Cf. A004613 (multiplicative closure).
Apart from initial term, same as A002313.
For values of n see A005098.
Primes in A020668.

a002144 n = a002144_list !! (n-1)
a002144_list = filter ((== 1) . a010051) [1,5..]
-- Reinhard Zumkeller, Mar 06 2012, Feb 22 2011

[a: n in [0..200] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Nov 23 2014

a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a),4*n+1]; fi; od: A002144 := n->a[n];
# alternative
A002144 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+4 by 4 do
            if isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002144(n),n=1..100) ; # R. J. Mathar, Jan 31 2024

Select[4*Range[140] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 16 2006 *)
Select[Prime[Range[150]],Mod[#,4]==1&] (* Harvey P. Dale, Jan 28 2021 *)

PARI
```
select(p->p%4==1,primes(1000))
```

A002144_next(p=A2144[#A2144])={until(isprime(p+=4),);p} /* NB: p must be of the form 4k+1. Beyond primelimit, this is *much* faster than forprime(p=...,, p%4==1 && return(p)). */
A2144=List(5); A002144(n)={while(#A2144A002144_next())); A2144[n]}
\\ M. F. Hasler, Jul 06 2024

from sympy import prime
A002144 = [n for n in (prime(x) for x in range(1,10**3)) if not (n-1) % 4]
# Chai Wah Wu, Sep 01 2014

from sympy import isprime
print(list(filter(isprime, range(1, 618, 4)))) # Michael S. Branicky, May 13 2021

def A002144_list(n): # returns all Pythagorean primes <= n
    return [x for x in prime_range(5,n+1) if x % 4 == 1]
A002144_list(617) # Peter Luschny, Sep 12 2012

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x < y) or of form u^2 + 4*v^2, (u = A002972, v = A002973, with u odd). - Lekraj Beedassy, Jul 16 2004
p^2 - 1 = 12*Sum_{i = 0..floor(p/4)} floor(sqrt(i*p)) where p = a(n) = 4*n + 1. [Shirali]
a(n) = A000290(A002972(n)) + A000290(2*A002973(n)) = A000290(A002331(n+1)) + A000290(A002330(n+1)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A002972(n)^2 + (2*A002973(n))^2, n >= 1. See the Jean-Christophe Hervé Nov 11 2013 comment. - Wolfdieter Lang, Jan 13 2015
a(n) = 4*A005098(n) + 1. - Zak Seidov, Sep 16 2018
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A088539.
Product_{k>=1} (1 + 1/a(k)^2) = A243380.
Product_{k>=1} (1 - 1/a(k)^3) = A334425.
Product_{k>=1} (1 + 1/a(k)^3) = A334424.
Product_{k>=1} (1 - 1/a(k)^4) = A334446.
Product_{k>=1} (1 + 1/a(k)^4) = A334445.
Product_{k>=1} (1 - 1/a(k)^5) = A334450.
Product_{k>=1} (1 + 1/a(k)^5) = A334449. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/A002145(k)) / (1 + 1/a(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/A002145(k)) / (1 - 1/a(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*n*s)! * zeta(n*s) * abs(EulerE(n*s - 1)) / (Pi^(n*s) * 2^(2*n*s) * BernoulliB(2*n*s) * (2^(n*s) + 1) * (n*s - 1)!))/n, s >= 3 odd number. - Dimitris Valianatos, May 21 2020
Legendre symbol (-1, a(n)) = +1, for n >= 1. - Wolfdieter Lang, Mar 03 2021

A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...

There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.

			1.0561821217268161417379307653162198905...

S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A004613, A004614, A013661, A175646.

digits = 105;
LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/  DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];
RealDigits[1/(4*LandauRamanujanK/Pi)^2, 10, digits][[1]] (* Jean-François Alcover, Jan 12 2021 *)

Equals 1/A088539. - Vaclav Kotesovec, May 05 2020
From Amiram Eldar, Sep 27 2020: (Start)
Equals Sum_{k>=1} 1/A004613(k)^2.
The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)

More digits from Vaclav Kotesovec, Jun 27 2020

A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 01 2020

The asymptotic density of the numbers k such that k^2+1 is squarefree (A049533) (Estermann, 1931).
The constant c in Sum_{k=0..n} phi(k^2 + 1) = A333170(n) ~ (1/4)*c*n^3 (Finch, 2018).
The constant c in Sum_{k=0..n} phi(k^2 + 1)/(k^2 + 1) = (3/4)*c*n + O(log(n)^2) (Postnikov, 1988).

			0.89484122456248817072566150690843732198754780892071...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 101.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 166.
A. G. Postnikov, Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, pp. 192-195.

Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Mathematische Annalen, Vol. 105 (1931), pp. 653-662, alternative link.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 14.
D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica, Vol. 155, No. 1 (2012), pp. 1-13.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], variable F(m=4,n=1,s=2), p. 38.
Wolfgang Schwarz, Über die Summe Sigma_{n <= x} phi(f)(n) und verwandte Probleme, Monatshefte für Mathematik, Vol. 66, No. 1 (1962), pp. 43-54, alternative link.
Radoslav Tsvetkov, On the distribution of k-free numbers and r-tuples of k-free numbers. A survey, Notes on Number Theory and Discrete Mathematics, Vol. 25, No. 3 (2019), pp. 207-222.

Cf. A002144, A049533, A065474, A069987, A088539, A333169, A333170, A340617.

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
pzetamod41 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(alfa((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
evalf(exp(-Sum(2^t*pzetamod41(2*t, 50)/t, t = 1..200))); # Vaclav Kotesovec, Jan 13 2021

f[p_] := If[Mod[p, 4] == 1, 1 - 2/p^2, 1]; RealDigits[N[Product[f[Prime[i]], {i, 1, 10^6}], 10], 10, 8][[1]] (* for calculating only the first few terms *)
(* -------------------------------------------------------------------------- *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

f(lim,poly=1-'x-'x^2/2)=prodeulerrat(subst(poly,'x,1/'x^2))*prodeuler(p=2,lim, my(pm2=1./p^2); if(p%4==1,1.-2*pm2,1.)/subst(poly,'x,pm2)) \\ Gets 14 digits at lim=1e9; Charles R Greathouse IV, Aug 10 2022

Equals 2*A065474/A340617.

More digits (from the paper by R. J. Mathar) added by Jon E. Schoenfield, Jan 12 2021

More digits from Vaclav Kotesovec, Jan 13 2021

A243379 Decimal expansion of 1/(2*K^2) = Product_(p prime congruent to 3 modulo 4) (1 - 1/p^2), where K is the Landau-Ramanujan constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 04 2014

Equals 1/1.168075586.., where 1.168.. is zeta_(m=4,n=3)(s=2) in the table of Section 3.3 of arxiv:1008.2547. - R. J. Mathar, Nov 14 2014

			0.856108981721893476906033006148061173481...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's MathWorld, Ramanujan constant

Cf. A002145, A064533, A088539, A243381.

digits = 101; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 1/(2*LandauRamanujanK^2) // RealDigits[#, 10, digits] & // First (* updated Mar 18 2018 *)

1/(2*K^2), where K is the Landau-Ramanujan constant (A064533).

A088539 * A243379 = 8 / Pi^2. - Vaclav Kotesovec, Apr 30 2020

A080109 Square of primes of the form 4k+1 (A002144).

Original entry on oeis.org

25, 169, 289, 841, 1369, 1681, 2809, 3721, 5329, 7921, 9409, 10201, 11881, 12769, 18769, 22201, 24649, 29929, 32761, 37249, 38809, 52441, 54289, 58081, 66049, 72361, 76729, 78961, 85849, 97969, 100489, 113569, 121801, 124609, 139129

Offset: 1

Cino Hilliard, Mar 16 2003

a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.
a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three...

			a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.
a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publ. No. 256, Vol II, Washington, DC, 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.

Cf. A002144, A080175. - Wolfdieter Lang, Jan 13 2015

Cf. A070079, A070151, A088539, A243380.

Select[4 Range[96] + 1, PrimeQ]^2 (* Michael De Vlieger, Dec 27 2016 *)

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^2" ")) ) }

a(n) = A002144(n)^2 = A070079(n)^2 + (4*A070151(n))^2, for n >= 1. - Wolfdieter Lang, Jan 13 2015
From Amiram Eldar, Dec 02 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A243380
Product_{n>=1} (1 - 1/a(n)) = A088539. (End)

Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015

A243380 Decimal expansion of 192K^2G/Pi^4 = Product_{p prime congruent to 1 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 04 2014

			1.0544399447999484896488194648267179483...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Eric Weisstein's MathWorld, Ramanujan constant.
Eric Weisstein's MathWorld, Catalan's Constant.

Cf. A002144, A064533, A088539.

digits = 101; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 192*LandauRamanujanK^2*Catalan/Pi^4 // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

Equals 192*K^2*G/Pi^4, where K is the Landau-Ramanujan constant (A064533) and G Catalan's constant (A006752).
A243380 * A243381 = 12/Pi^2. - Vaclav Kotesovec, Apr 30 2020
Equals A175647 / 1.001652229636651... both constants from p 26 of arXiv:1008.2537v2. - R. J. Mathar, Aug 21 2022

A334425 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Apr 30 2020

			0.991251116234099844239776364609097744339415...

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 1 3 = 1/A334425).

Cf. A002144, A088539, A334446, A334450.

A334424 / A334425 = 105*zeta(3)/(4*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

A334450 Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Apr 30 2020

In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - Dimitris Valianatos, May 01 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)).

			0.999676527079626662018246180873083701500751574379554430568432840424975981921219...

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 1 5 = 1/A334450).

Cf. A002144, A088539, A334446, A334450.

A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

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

More digits from Vaclav Kotesovec, Jun 27 2020

A096018 Number of Pythagorean quadruples mod n; i.e., number of solutions to w^2 + x^2 + y^2 = z^2 mod n.

Original entry on oeis.org

1, 8, 21, 64, 145, 168, 301, 512, 621, 1160, 1221, 1344, 2353, 2408, 3045, 4096, 5185, 4968, 6517, 9280, 6321, 9768, 11661, 10752, 18625, 18824, 16281, 19264, 25201, 24360, 28861, 32768, 25641, 41480, 43645, 39744, 51985, 52136, 49413, 74240

Offset: 1

T. D. Noe, Jun 15 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..127 from R. J. Mathar, terms 128..2500 from Andrew Howroyd)
A. H. Hakami, Small zeros of quadratic congruences to a prime power modulus, PhD Thesis (2009), Lemma 4.4.
A. H. Hakami, Small primitive zeros of quadratic forms mod p^m, Raman. J. 38 (2015) 189-198, Lemma 2.1 for n=4, det Q=-1, omega_j(y')= p^(m-j)-p^(m-j-1).
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.
Index to sequences related to sums of squares.

Cf. A062775 (number of solutions to x^2 + y^2 = z^2 mod n), A240547.

Cf. A088539, A243381, A334425, A334426.

A096018 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(3*e) ;
        elif modp(p,4) = 1 then
            a := a* p^(2*e-1)*(p^(e+1)+p^e-1) ;
        else
            if type(e,'even') then
                a := a* (p^(3*e)+(p-1)*p^(2*e-1)*(1-p^e)/(1+p)) ;
            else
                a := a* (p^(3*e)-(p-1)*p^(2*e-1)*(1+p^e)/(1+p)) ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A096018(n),n=1..50) ; # R. J. Mathar, Jun 24 2018

Table[cnt=0; Do[If[Mod[w^2+x^2+y^2-z^2, n]==0, cnt++ ], {w, 0, n-1}, {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; cnt, {n, 50}]
f[2, e_] := 2^(3*e); f[p_, e_] := If[Mod[p, 4] == 1, p^(2*e - 1)*(p^(e + 1) + p^e - 1), If[EvenQ[e], p^(3*e) + (p - 1)*p^(2*e - 1)*(1 - p^e)/(1 + p), p^(3*e) - (p - 1)*p^(2*e - 1)*(1 + p^e)/(1 + p)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

M(n,f)={sum(i=0, n-1, Mod(x^(f(i)%n), x^n-1))}
a(n)={polcoeff(lift(M(n, i->i^2)^3 * M(n, i->-(i^2))), 0)} \\ Andrew Howroyd, Jun 23 2018

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, 2^(3*e), if(p%4 == 1, p^(2*e-1)*(p^(e+1) + p^e - 1), if(e%2, p^(3*e) - (p - 1)*p^(2*e - 1)*(1 + p^e)/(1 + p), p^(3*e) + (p - 1)*p^(2*e - 1)*(1 - p^e)/(1 + p)))));} \\ Amiram Eldar, Nov 21 2023

a(n) is multiplicative. For the powers of primes p, there are several cases. For p=2, we have a(2^e) = 2^(3e). For odd primes p with p==1 (mod 4), we have a(p^e) = p^(2*e-1)*(p^(e+1)+p^e-1). For odd primes p with p==3 (mod 4) and even e we have a(p^e) = p^(3*e) +(p-1)*p^(2*e-1)*(1-p^e)/(1+p). For odd primes p == 3 (mod 4) and odd e we have a(p^e) = p^(3*e) -(p-1)*p^(2*e-1)*(1+p^e)/(1+p). [Corrected Jun 24 2018, R. J. Mathar]

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A334425 * A334426 /(A088539 * A243381) = 0.94532146880744347512... . - Amiram Eldar, Nov 21 2023

cp's OEIS Frontend

A002144 Pythagorean primes: primes of the form 4*k + 1.

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A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

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A335963 Decimal expansion of Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2).

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A243379 Decimal expansion of 1/(2*K^2) = Product_(p prime congruent to 3 modulo 4) (1 - 1/p^2), where K is the Landau-Ramanujan constant.

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A080109 Square of primes of the form 4k+1 (A002144).

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A243380 Decimal expansion of 192*K^2*G/Pi^4 = Product_{p prime congruent to 1 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.

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A243380 Decimal expansion of 192K^2G/Pi^4 = Product_{p prime congruent to 1 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.