A175647 -id:A175647 - cp's OEIS frontend

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.

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

A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.273986613206833925158...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler product, arXiv:1908.06808 [math.NT], 2019, p. 20.
Steven Finch, Quartic and Octic Characters Modulo n, arXiv:1008.2547 [math.NT], 2007-2010 p. 11 (formula on kappa(5) and kappa(-5)).
Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (formulas).
Alessandro Languasco and Alessandro Zaccagnini, A note on Mertens' formula for arithmetic progressions, Journal of Number Theory Volume 127, Issue 1, (2007), 37-46.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data).
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
G. Niklasch, Some number-theoretical constants arising as products of rational functions of p over primes
Doug S. Phillips and Peter Zvengrowski, Convergence of Dirichlet Series and Euler Products, Contributions Section of Natural Mathematical and Biotechnical Sciences 38(2):153 (2017).

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340710.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 3, 4]/Z[5, 3, 2]^2)]

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = A340710.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = this constant.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Equals Sum_{q in A004617} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

Original entry on oeis.org

1, 3, 7, 11, 19, 21, 23, 31, 33, 43, 47, 57, 59, 67, 69, 71, 77, 79, 83, 93, 103, 107, 127, 129, 131, 133, 139, 141, 151, 161, 163, 167, 177, 179, 191, 199, 201, 209, 211, 213, 217, 223, 227, 231, 237, 239, 249, 251, 253, 263, 271, 283, 301, 307, 309, 311, 321, 329

Offset: 1

Arnaud Vernier, Oct 29 2009

Or, numbers that are not divisible by the sum of two squares (other than 1). - Clarified by Gabriel Conant, Apr 18 2016

If a term divides the sum of two squares, then it divides each of the two numbers individually. Moreover, only the numbers in this sequence have this property. See link for proof. - V Sai Prabhav, Jul 15 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
V Sai Prabhav, Proof for the comment

Cf. A002145, A004613, A004614, A005117, A231754.

Cf. A175647, A243379.

N:= 1000: # to get all terms <= N
S:= {1};
for p from 3 by 4 to N do
  if isprime(p) then
    S:= S union select(`<=`, map(t -> t*p, S),N)
  fi
od:
sort(convert(S,list)); # Robert Israel, Apr 18 2016

Select[Range@ 1000, #==1 || ({{3}, {1}} == Union /@ {Mod[ #[[1]], 4], #[[2]]} &@ Transpose@ FactorInteger@ #) &] (* Giovanni Resta, Apr 18 2016 *)

isok(n) = if (! issquarefree(n), return (0)); f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 3, return (0))); 1 \\ Michel Marcus, Sep 04 2013

A005117 INTERSECT A004614. - R. J. Mathar, Nov 05 2009

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A243379/(2*sqrt(A175647)) = 0.4165140462... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Edited by Zak Seidov, Oct 30 2009

Narrowed definition down to squarefree numbers - R. J. Mathar, Nov 05 2009

A340665 Decimal expansion of Product_{primes p == 3 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 15 2021

			1.135764878668921626868643009472082289511936413...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions..., arXiv:1008.2547 Zeta_{m=5,n=3}(s=2).
For links see A340628.

Cf. A004617, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340628, A340004, A340127.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 100; digits = 50; (* Adjust as needed. *)
digitize[c_] := RealDigits[Chop[N[c, digits+10]], 10, digits][[1]];
digitize[Z[5, 3, 2]]

Equals Sum_{k>=1} 1/A004617(k)^2. - Amiram Eldar, Jan 24 2021

A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 21 2021

			1.36857205387664908586076389048310999017020782888589520500850402955633118881...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
For links see A340711.

Cf. A004616, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340127, A340576, A340577, A340578, A340628, A340629, A340665, A340710, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[Z[5, 2, 2]]

I = Product_{primes p == 0 (mod 5)} p^2/(p^2-1) = 25/24.
J = Product_{primes p == 1 (mod 5)} p^2/(p^2-1) = A340004.
K = Product_{primes p == 2 (mod 5)} p^2/(p^2-1) = this constant.
L = Product_{primes p == 3 (mod 5)} p^2/(p^2-1) = A340665.
M = Product_{primes p == 4 (mod 5)} p^2/(p^2-1) = A340127.
I*J*K*L*M = Pi^2/6 = zeta(2).
J*K*L*M = 4*Pi^2/25.
M = (Pi/2)*C(5,4)^(-2)*exp(-gamma/2)*sqrt(3/log(2+sqrt(5))), where gamma is the Euler-Mascheroni constant A001620 and C(5,4) is the Mertens constant = 1.29936454791497798816084...
Equals Sum_{k>=1} 1/A004616(k)^2. - Amiram Eldar, Jan 24 2021

A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 05 2011

Arises in studying A002496.

The constant is Product_{primes p} (1-chi(p)/(p-1)) where chi is the Dirichlet character A101455. Its Euler expansion is (1/(L(m=4,r=2,s=1)* zeta(m=4,n=3,s=2)) *Product_{s>=2} zeta(m=4,n=1,s)^gamma(s), where L and zeta are the functions tabulated in arXiv:1008.2547 and gamma is the sequence A001037. In particular L(m=4,r=2,s=1) = A003881 and zeta(m=4,n=1,s=2)=A175647. - R. J. Mathar, Nov 29 2011

			1.372813462818246009112192696727...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

T. Amdeberhan, L. A. Median, and V. H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory 128 (2008) 1807-1846, eq. (1.10).
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70. See Section 5.41.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496.
Cf. A001037, A003881, A083844, A175647, A221712, A331942.
Equals 2*constant given by A331941.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1) after setting the required precision.

Extended title, a(30) and beyond from Hugo Pfoertner, Feb 16 2020

A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 16 2021

			1.7551738411687377766074721228405237...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
For links see A340711.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340711.

(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 2, 4]/Z[5, 2, 2]^2)]

D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = this constant.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = A340711.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Formulas by Pascal Sebah, Jan 20 2021: (Start)
Let g = sqrt(Cl2(2*Pi/5)^2+Cl2(4*Pi/5)^2) = 1.0841621352693895..., where Cl2 is the Clausen function of order 2.
E = 15*sqrt(65)*g/(13*Pi^2).
H = 6*sqrt(13)*Pi^2/(195*g). (End)
Equals Sum_{q in A004616} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

A369563 Powerful numbers whose prime factors are all of the form 4*k + 1.

Original entry on oeis.org

1, 25, 125, 169, 289, 625, 841, 1369, 1681, 2197, 2809, 3125, 3721, 4225, 4913, 5329, 7225, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 21025, 21125, 22201, 24389, 24649, 28561, 29929, 32761, 34225, 36125, 37249, 38809, 42025, 48841, 50653, 52441, 54289, 54925

Offset: 1

Amiram Eldar, Jan 26 2024

nonn

Closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A004613.
Subsequence: A146945.
Similar sequence: A352492, A369564, A369565, A369566.
Cf. A002144, A008846, A175647, A334424.

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 1 && Last[#] > 1 &]; Select[Range[50000], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 1 || f[i, 2] == 1, return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) = A175647 * A334424 = 1.0654356335... .

A031358 Number of coincidence site lattices of index 4n+1 in lattice Z^2.

Original entry on oeis.org

1, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 4, 2, 0, 2, 0, 0, 2, 2, 0, 2, 4, 0, 2, 2, 0, 4, 0, 0, 0, 4, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 4, 2, 0, 2, 2, 0, 2, 2, 0, 0, 4, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 0, 2, 4, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2

Offset: 0

N. J. A. Sloane

Andrey Zabolotskiy, Table of n, a(n) for n = 0..999
M. Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See annotated scan of page 713.

Cf. A175647, A031359, A331140, A106594, A094178 (positions of nonzero terms).

t1=direuler(p=2,1200,(1+(p%4<2)*X))
t2=direuler(p=2,1200,1/(1-(p%4<2)*X))
t3=dirmul(t1,t2)
t4=vector(200,n,t3[4*n+1]) \\ and then prepend 1

Dirichlet series: Product_{primes p == 1 mod 4} (1+p^(-s))/(1-p^(-s)).

a(n) = 2*A106594(n) for n > 0. - Andrey Zabolotskiy, Jan 30 2020

More terms from N. J. A. Sloane, Mar 13 2009
Added condition that p must be prime to the Dirichlet series. - N. J. A. Sloane, May 26 2014
Offset corrected by Andrey Zabolotskiy, Jan 30 2020

A231754 Products of distinct primes congruent to 1 modulo 4 (A002144).

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 293, 305, 313, 317, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 433, 445, 449

Offset: 1

Michel Marcus, Nov 13 2013

Contains A002144 as a subsequence, and is a subsequence of A016813 and of A005117.

Also, these numbers satisfy A231589(n) = floor(n*(n-1)/4) (A011848).

			65 = 5*13 is in the sequence since both 5 and 13 are congruent to 1 modulo 4.

R. J. Mathar, Table of n, a(n) for n = 1..4999
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.
Shailesh A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag., Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Intersection of A005117 and A004613.
Cf. A002144, A011848, A016813, A167181, A231589.
Cf. A088539, A175647.

isA231754 := proc(n)
    local d;
    for d in ifactors(n)[2] do
        if op(2,d) > 1 then
            return false;
        elif modp(op(1,d),4) <> 1 then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 500 do
    if isA231754(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 16 2016

Select[Range[500], # == 1 || AllTrue[FactorInteger[#], Last[#1] == 1 && Mod[First[#1], 4] == 1 &] &] (* Amiram Eldar, Mar 08 2024 *)

isok(n) = if (! issquarefree(n), return (0)); if (n > 1, f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 1, return (0)))); 1

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A088539 * sqrt(A175647) / Pi = 0.3097281805... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

cp's OEIS Frontend

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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A340711 Decimal expansion of Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1).

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A167181 Squarefree numbers such that all prime factors are == 3 mod 4.

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A340665 Decimal expansion of Product_{primes p == 3 (mod 5)} p^2/(p^2-1).

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A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

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A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

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A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

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A369563 Powerful numbers whose prime factors are all of the form 4*k + 1.

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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.