A243381 -id:A243381 - 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

A050469 a(n) = Sum_{ d divides n, n/d=1 mod 4} d - Sum_{ d divides n, n/d=3 mod 4} d.

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 6, 8, 7, 12, 10, 8, 14, 12, 12, 16, 18, 14, 18, 24, 12, 20, 22, 16, 31, 28, 20, 24, 30, 24, 30, 32, 20, 36, 36, 28, 38, 36, 28, 48, 42, 24, 42, 40, 42, 44, 46, 32, 43, 62, 36, 56, 54, 40, 60, 48, 36, 60, 58, 48, 62, 60, 42, 64, 84, 40

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative with a(p^e)=p^e if p=2, (p^(e+1)-1)/(p-1) if p==1 (mod 4), else (p^(e+1)+(-1)^e)/(p+1). - Michael Somos, May 02 2005

Multiplicative because it is the Dirichlet convolution of A000027 = n and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A006752, A050470, A050471, A050468, A175647, A243381.

max = 70; s = Sum[n*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015 *)
f[p_, e_] := Which[p == 2, p^e, Mod[p, 4] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 4] == 3, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 06 2022 *)

a(n)=if(n<1,0,sumdiv(n,d,d*((n/d%4==1)-(n/d%4==3))))

{a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, p^e, if(p%4==1, (p^(e+1)-1)/(p-1), (p^(e+1)+(-1)^e)/(p+1)))))) } /* Michael Somos, May 02 2005 */

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(1+x^(2*k)),x*O(x^n)),n))

G.f.: Sum_{n>=1} n*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
L.g.f.: Sum_{k>=1} arctan(x^k). - Ilya Gutkovskiy, Dec 16 2019
O.g.f.: Sum_{n >= 1} (-1)^(n+1) * x^(2*n-1)/(1 - x^(2*n-1))^2. - Peter Bala, Jan 04 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 4)} 1/(1-1/p^2) * Product_{primes p == 3 (mod 4)} 1/(1+1/p^2) = (1/2) * A175647 / A243381 = A006752/2 = 0.4579827970... . - Amiram Eldar, Nov 06 2022, Nov 05 2023
a(n) = Sum_{d|n} (n/d)*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

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

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

A334426 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 30 2020

			1.041158072823444580338360569925615669376071...

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, A334447, A334451.

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

A334424 * A334426 = 840*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

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

A334451 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^5).

Original entry on oeis.org

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

Offset: 1

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)).

			1.0041818167708566903887269765658569606315819506367432882834249768697794496439...

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, A334447.

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

A334449 * A334451 = 90720*zeta(5)/Pi^10.

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

A242822 Decimal expansion of B. Davis' constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 23 2014

			1.3468852519994065951820075554411...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001221, A002145, A004614, A006752, A111003, A330890, A377753.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/(8*Catalan(R)); // G. C. Greubel, Aug 25 2018

s:= convert(evalf(Pi^2/(8*Catalan), 140), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

RealDigits[Pi^2/(8*Catalan), 10, 100] // First

default(realprecision, 100); Pi^2/(8*Catalan) \\ G. C. Greubel, Aug 25 2018

(Sum_{n>=0} 1/(2*n + 1)^2) / (Sum_{n>=0} (-1)^n/(2*n + 1)^2) = A111003/A006752.
Equals Product_{k>=1} (1 + 1/A002145(k)^2)/(1 - 1/A002145(k)^2) = A243381 / A243379. - Vaclav Kotesovec, Apr 30 2020
Equals Sum_{q in A004614} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021
Equals 1/A377753. - Hugo Pfoertner, Nov 22 2024

A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 4).

Original entry on oeis.org

1, 1, 4, 2, 4, 4, 8, 4, 12, 4, 12, 8, 12, 8, 16, 8, 16, 12, 20, 8, 32, 12, 24, 16, 20, 12, 36, 16, 28, 16, 32, 16, 48, 16, 32, 24, 36, 20, 48, 16, 40, 32, 44, 24, 48, 24, 48, 32, 56, 20, 64, 24, 52, 36, 48, 32, 80, 28, 60, 32, 60, 32, 96, 32, 48, 48, 68, 32

Offset: 1

José María Grau Ribas, Jan 17 2012

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A072437.
Cf. A175647, A243381.
Cf. A079458, A062570.

with(numtheory):
a := n->add(jacobi(-1,d)*mobius(d)*n/d, d in divisors(n)):
seq(a(n), n = 1..60); # Peter Bala, Dec 26 2023

ar[p_,s_] := Which[Mod[p,4]==1, p^(s-1)*(p-1), Mod[p,4]==3, p^(s-1)*(p+1), True,p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]

A204617(n) = { my(f=factor(n),p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p,1,if(1==(p%4),p-1,p+1))); }; \\ Antti Karttunen, Nov 16 2021

a(n) = phi(n) if n is in A072437.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - Amiram Eldar, Dec 24 2022
a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - Peter Bala, Dec 26 2023
a(n) = A079458(n)/A062570(n). - Ridouane Oudra, Jun 04 2024

cp's OEIS Frontend

A002145 Primes of the form 4*k + 3.

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A050469 a(n) = Sum_{ d divides n, n/d=1 mod 4} d - Sum_{ d divides n, n/d=3 mod 4} d.

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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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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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A334426 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^3).

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

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A334451 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^5).

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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.