A064533 -id:A064533 - cp's OEIS frontend

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 146, 149, 157, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 305, 313, 314, 317, 337, 346, 349

Offset: 1

Steven Finch

nonn

Primitively but not imprimitively represented by x^2 + y^2.
The disjoint union of {1}, A003654, and A031398. - Max Alekseyev, Mar 09 2010
Squarefree members of A202057. - Artur Jasinski, Dec 10 2011
Union of A231754 and 2*A231754. Squarefree numbers whose prime factors are in A002313. - Robert Israel, Aug 23 2017
It appears that a(n) is the n-th index, k, such that f(k) = 2, where f(k) = 3*(Sum_{i=1..k} floor(i^2/k)) - k^2 (see A175908). - John W. Layman, May 16 2011

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Index entries for sequences related to sums of squares

Cf. A001481, A008784, A022544, A034023, A175908.

Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

a020893 n = a020893_list !! (n-1)
a020893_list = filter (\x -> any (== 1) $ map (a010052 . (x -)) $
                             takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, May 28 2015

N:= 1000: # to get all terms <= N
R:= {1,2}:
p:= 2:
do
p:= nextprime(p);
if p > N then break fi;
if p mod 4 <> 1 then next fi;
R:= R union select(`<=`,map(`*`,R,p),N);
od:
sort(convert(R,list)); # Robert Israel, Aug 23 2017

lim = 17; t = Join[{1}, Select[Union[Flatten[Table[x^2 + y^2, {x, lim}, {y, x}]]], # < lim^2 && SquareFreeQ[#] &]]
Select[Union[Total/@Tuples[Range[0,20]^2,2]],SquareFreeQ] (* Harvey P. Dale, Jul 26 2017 *)
Block[{nn = 350, p}, p = {1, 2}~Join~Select[Prime@ Range@ PrimePi@ nn, Mod[#, 4] == 1 &]; Select[Range@ nn, And[SquareFreeQ@ #, SubsetQ[p, FactorInteger[#][[All, 1]]]] &]] (* Michael De Vlieger, Aug 23 2017 *)
(* or *)
Select[Range[350], SquareFreeQ@ # && ! MemberQ[Mod[First /@ FactorInteger@ #, 4], 3] &] (* Giovanni Resta, Aug 25 2017 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 || f[i,1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015

from itertools import count, islice
from sympy import factorint
def A020893_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 and e == 1 for p, e in factorint(n).items()),count(1))
A020893_list = list(islice(A020893_gen(),30)) # Chai Wah Wu, Jun 28 2022

a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015

Edited by N. J. A. Sloane, Aug 30 2017

A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

Original entry on oeis.org

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

Offset: 0

Michel Waldschmidt, Mar 21 2018

This is the decimal expansion of the number alpha such that the number of positive integers <= N which are sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4).

Based on the constants Zeta(m=12,n=5,s=2) = 1.0482019036007..., Zeta(m=12,n=7,s=2) = 1.0262021468... and Zeta(m=12,n=11,s=2) = 1.01177863 ... read from arXiv:1008.2547 we have Product_{p == 5, 7, 11(mod 12)} (1-1/p^2)^(-1/2) = sqrt( Zeta(m=12,n=5,s=2) * Zeta(m=12,n=7,s=2) * Zeta(m=12,n=11,s=2) ) as a factor in the formulas. - R. J. Mathar, Feb 04 2021

			0.30231614235706563794776990048019971560241279...

Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 17.
Olivier Ramaré, S. Ettahri, and L. Surel, Fast multi-precision computation of some Euler products, Mathematics of Computation (2021) hal-03381427.

Cf. A003136, A064533, A167135, A301429, A340552.

Digits:= 1000: with(numtheory):
B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))):
for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B;

prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt[3]])/2)^(1/4))/(2 Gamma[1/4]), prec];
For[n = 3, n < 50000, n++, p = Prime[n];
If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]]
Print[B] (* Peter Luschny, Mar 23 2018 *)
(* -------------------------------------------------------------------------- *)
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}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[(3^(1/4)/2^(5/4)) * Pi^(1/2) * (Log[2 + Sqrt[3]])^(1/4) / Gamma[1/4] * Sqrt[Z[12, 5, 2] * Z[12, 7, 2] * Z[12, 11, 2]], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2).

One can base the definition on p(n) = A167135(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018

Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jan 15 2021

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

A160499 Number of quartic primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 1, 2, 0, 0, 1, 1, 3, 0, 3, 4, 3, 0, 1, 3, 1, 0, 1, 2, 0, 0, 0, 1, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 3, 6, 3, 0, 1, 1, 0, 0, 1, 4, 0, 0, 3, 3, 3, 0, 3, 2, 1, 0, 1, 3, 3, 0, 0, 0, 9, 0, 1, 3, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 12, 0

Offset: 1

Steven Finch, May 15 2009

Also called biquadratic primitive Dirichlet characters.
Primitive Dirichlet characters of both order 2 & order 4 are included.
a(n) is the number of primitive Dirichlet characters modulo n such that all entries are 0 or a fourth-power root of unity (1, i, -1 and -i). - Jianing Song, Feb 27 2019
Mobius transform of A073103. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Mar 02 2019: (Start)
For n = 5, the 3 quartic primitive Dirichlet characters modulo n are [0, 1, -1, -1, 1], [0, 1, i, -i, -1] and [0, 1, -i, i, -1], so a(5) = 3.
For n = 16, the 4 quartic primitive Dirichlet characters modulo n are [0, 1, 0, i, 0, i, 0, 1, 0, -1, 0, -i, 0, -i, 0, -1], [0, 1, 0, -i, 0, -i, 0, 1, 0, -1, 0, i, 0, i, 0, -1], [0, 1, 0, i, 0, -i, 0, -1, 0, -1, 0, -i, 0, i, 0, 1] and [0, 1, 0, -i, 0, i, 0, -1, 0, -1, 0, i, 0, -i, 0, 1], so a(16) = 4. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Cubic and quartic characters.
Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2016.

Cf. A114643 (number of quadratic primitive Dirichlet characters modulo n), A160498 (number of cubic primitive Dirichlet characters modulo n).
Cf. A073103 (number of solutions to x^4 == 1 (mod n)).
Cf. A064533.

f[n_] := Sum[If[Mod[k^4 - 1, n] == 0, 1, 0], {k, 1, n}]; a[n_] := Sum[ MoebiusMu[n/d]*f[d], {d, Divisors[n]}]; Table[a[n], {n, 2, 81}] (* Jean-François Alcover, Jun 19 2013 *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e == 4, 4, e >= 5, 0]; f[p_, 1] := If[Mod[p, 4] == 1, 3, 1]; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^4-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

Multiplicative with a(4) = 1, a(8) = 2, a(16) = 4, a(2^e) = 0 for e = 1 or e >= 5; for odd primes p, a(p) = 3 if p == 1 (mod 4) and 1 if p == 3 (mod 4), a(p^e) = 0 for e >= 2. - Jianing Song, Mar 02 2019

Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (7/(16*Pi*K^2)) * Product_{primes p == 1 (mod 4)} (1 - (5*p-3)/(p^2*(p+1))) = 0.1908767211685284480112237..., and K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Sep 16 2020

a(1) = 1 prepended by Jianing Song, Feb 27 2019

A227158 Second-order term in the asymptotic expansion of B(x), the count of numbers up to x which are the sum of two squares.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Jul 03 2013

K = A064533, the Landau-Ramanujan constant, is the first-order term. This constant is c = lim_{x->oo} (B(x)*sqrt(log x)/(K*x) - 1)*log x. [Corrected by Alessandro Languasco, Sep 14 2022]

130000 digits are available, see link to web page. - Alessandro Languasco, Mar 27 2024

			0.58194865931729079777136487517474826173838317235153574360562...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.

Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024. See p. 32.
Alexandru Ciolan, Alessandro Languasco and Pieter Moree, Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms, section 10, 47500 digits are obtained, Journal of Mathematical Analysis and Applications, 2022; see also preprint on arXiv, arXiv:2109.03288 [math.NT], 2021.
Alessandro Languasco, Programs and numerical results, providing 130000 digits. [Note: information ancillary to above link.]
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 9.
David Hare, Landau-Ramanujan Constant, second order obtained about 5000 digits, 1996.
Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.

Cf. A064533, A001481.

digits = 101; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[ Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10] ; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; RealDigits[f[m], 10, digits] // First (* Jean-François Alcover, May 27 2014 *)

L(s)=sumalt(k=0,(-1)^k/(2*k+1)^s)
LL(s)=L'(s)/L(s)
ZZ(s)=zeta'(s)/zeta(s)
sm(x)=my(s);forprime(q=2,x,if(q%4==3,s+=log(q)/(q^8-1))); s+1/49/x^7+log(x)/7/x^7
1/2+log(2)/4-Euler/4-LL(1)/4-ZZ(2)/4+LL(2)/4-log(2)/12-ZZ(4)/4+LL(4)/4-log(2)/60+sm(1e5)/2

Corrected and extended by Jean-François Alcover, Mar 19 2014 and again May 27 2014

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Nov 16 2003

An illustration of the Chebyshev effect.

			1.2923041571286886071...

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 5.
S. Uchiyama, On some products involving primes, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

Cf. A087197, A064533, A088541, A155739.

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]; 4/Sqrt[Pi]*Exp[-EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Jun 04 2014, updated Mar 14 2018 *)

Equals (4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).

Equals 4*A087197*A064533/exp(A155739). - R. J. Mathar, Feb 05 2009

Offset corrected by R. J. Mathar, Feb 05 2009

A000690 Landau's approximation to population of x^2 + y^2 <= 2^n.

Original entry on oeis.org

1, 2, 3, 4, 7, 13, 24, 44, 83, 157, 297, 567, 1085, 2086, 4019, 7766, 15039, 29181, 56717, 110408, 215225, 420076, 820836, 1605587, 3143562, 6160098, 12080946, 23710229, 46565965, 91512121, 179947985, 354043613, 696935548, 1372589372

Offset: 0

N. J. A. Sloane

nonn

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).

Sean A. Irvine, Table of n, a(n) for n = 0..999
D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Index entries for sequences related to populations of quadratic forms

Cf. A000050, A064533.

a(n) = round(b*2^n/sqrt(log(2^n))) where b=0.764223654... is the Landau-Ramanujan constant (A064533).

More terms from Sean A. Irvine, Feb 23 2011

Name clarified by Seth A. Troisi, Apr 28 2022

A000691 Ramanujan's approximation to population of x^2 + y^2 <= 2^n.

Original entry on oeis.org

1, 2, 3, 5, 9, 16, 29, 52, 94, 175, 327, 616, 1169, 2231, 4273, 8215, 15842, 30628, 59345, 115208, 224040, 436343, 850981, 1661663, 3248231, 6356076, 12448925, 24402959, 47873156, 93984236, 184632691, 362938014, 713852252, 1404817026

Offset: 0

N. J. A. Sloane

nonn

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).

D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Index entries for sequences related to populations of quadratic forms

K = A064533.

Other population sequences for x^2 + y^2: A000050, A000690, A000692.

Digits:=500;
K:=.764223653589220662990698731250092328116790541393409514721686673
7496146416587328588384015050131312337219372691207925926341874206467
8084323063315434629380531605171169636177508819961243824994277683469
0516235139218719620569053295644670419176349770659569905712938660289
3858998296105166296089099177929836072973697200640316985128636517347
3921065768550978681981674707359066921; a:=n->round(evalf(K*int(1/sqrt(ln(t)),t=1..2^n))); # Salvador Perez (pies314(AT)hotmail.com), May 08 2005

More terms from Salvador Perez (pies314(AT)hotmail.com), May 08 2005
Corrected by Sean A. Irvine, Feb 24 2011
Name clarified by Seth A. Troisi, May 23 2022

A053443 x^2 + y^2 does not take on all possible values mod n.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 126, 128, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171

Offset: 1

David W. Wilson

nonn

Sequence gives values of n such there is not always a solution 1 < z < n to x^2 + y^2 = z (mod n). - Benoit Cloitre, Jan 04 2002; corrected by Carmine Suriano, Jun 19 2013

The asymptotic density of this sequence is 1- 3/(8*K^2) = 1 - (3/4) * A243379 = 0.35791..., where K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Dec 19 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A240370.

Cf. A053444, A064533, A243379.

Select[Range[200], AnyTrue[FactorInteger[#], Mod[First[#1], 4] > 1 && Last[#1] > 1 &] &] (* Amiram Eldar, Dec 19 2020 *)

is(n)=my(v=vectorsmall(n,i,1));for(x=0,n\2, for(y=0,x, v[(x^2+y^2)%n+1]=0)); vecmax(v) \\ Charles R Greathouse IV, Jun 19 2013

is(n)=forprime(p=2,97,my(o=valuation(n,p));if(o,if(o>1&&p%4>1,return(1));n/=p^o));my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1&&f[i,1]%4>1,return(1)));0 \\ Charles R Greathouse IV, Jun 19 2013

n divisible by p^2 where p = 2 or prime p == 3 (mod 4).

A075880 Position of the circles around (0,0) that contain record numbers of lattice points in the list of all circles around (0,0) that pass through lattice points, ordered by increasing radius.

Original entry on oeis.org

0, 1, 4, 13, 30, 121, 362, 1232, 1584, 7121, 17548, 32649, 37603, 174926, 437750, 821432, 1198677, 5678338, 14335447, 27044791, 43735981, 209473053, 531787054, 1006745669, 2097411347, 8474384496, 10122355701

Offset: 1

Hugo Pfoertner, Oct 16 2002

nonn

The first terms of this sequence were given by James Buddenhagen in a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points"

			a(4)=13 because A001481(14) = A071383(4)=25.

James Buddenhagen, Circle with 3 lattice points, thread in NG sci.math.
Hugo Pfoertner, Asymptotic Behavior of A075880(n)

Cf. A001481, A064533, A071383, A230656.

a(n) = k-1 for which A001481(k) = A071383(n).

lim n ->infinity a(n) = k_LR * exp(n) / n^(1/2), where k_LR is the Landau-Ramanujan constant 0.764223653... (A064533)

Minor edits to adjust formula and example for changes to offset of related sequences by Ray Chandler, Jan 13 2012

cp's OEIS Frontend

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

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A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

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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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A160499 Number of quartic primitive Dirichlet characters modulo n.

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A227158 Second-order term in the asymptotic expansion of B(x), the count of numbers up to x which are the sum of two squares.

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A088540 Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni 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.

A088540 Decimal expansion of (4/sqrt(Pi))exp(-gamma/2)K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.