A000050 -id:A000050 - cp's OEIS frontend

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

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

A000074 Number of odd integers <= 2^n of form x^2 + y^2.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 83, 157, 296, 564, 1083, 2077, 4006, 7733, 14968, 29044, 56447, 109864, 214197, 418080, 816907, 1598040, 3129063, 6132106, 12027122, 23606527, 46366165, 91127332, 179207074, 352615528, 694182554, 1367278759

Offset: 1

N. J. A. Sloane

nonn

First differences of A000050. - Jean-François Alcover, Mar 19 2014

			a(4)=4 since 2^4=16 and 1=1^2, 5=1^2+2^2, 9=3^2, 13=2^2+3^2.

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).

Seth A. Troisi, Table of n, a(n) for n = 1..50
Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Math. Comp., 18 (1964), 75-86.
Index entries for sequences related to populations of quadratic forms

Cf. A000050.

PARI
```
a(n)=if(n<0,0,sum(k=1,2^(n-1),0
				
```

9 more terms from Sean A. Irvine, Sep 14 2009

A000692 An approximation to population of x^2 + y^2 <= 2^n.

Original entry on oeis.org

1, 3, 4, 5, 9, 15, 27, 50, 92, 171, 322, 610, 1161, 2220, 4260, 8201, 15828, 30622, 59362, 115287, 224260, 436871, 852161, 1664196, 3253531, 6366973, 12471056, 24447507, 47962236, 94161474, 184983976, 363632192, 715220838, 1407510311

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. Hare, The constant c [Dave Hare, May 21 1996].
D. Shanks, The second-order term in the asymptotic expansion of B(x), Math. Comp., 18 (1964), 75-86.
Index entries for sequences related to populations of quadratic forms.

Cf. A064533.

Other population sequences for x^2 + y^2: A000050, A000690, A000691.

a(n) = (b*2^n / sqrt(n*log(2))) * (1 + c/(n*log(2))) where b=0.764223654... is the Landau-Ramanujan constant (A064533) and c=0.5819486593... is the second-order Landau-Ramanujan constant (A227158) given by c = (1/2) * (1-log(Pi*e^gamma/(2*L))) - (1/4) * D(1) where D(s) = (d/ds)(log(Product_{p prime == 3 (mod 4)} 1/(1-p^(-2*s)))) and L is the Lemniscate constant (A064853) [see (12) in Shanks]. - Sean A. Irvine, Feb 25 2011

More terms from Sean A. Irvine, Feb 24 2011

Name clarified by Seth A. Troisi, May 23 2022

A164775 a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

Original entry on oeis.org

7, 43, 330, 2749, 24028, 216341, 1985459, 18457847, 173229058, 1637624156, 15570512744, 148736628858, 1426306930865, 13722217893214, 132387263219058, 1280309691127436

Offset: 1

Eric W. Weisstein, Aug 26 2009

			a(1)=7 since 1 = 0^2 + 1^2, 2 = 1^2 + 1^2, 4 = 0^2 + 2^2, 5 = 1^2 + 2^2, 8 = 2^2 + 2^2, 9 = 0^2 + 3^2, 10 = 1^2 + 3^3.

Peter Shiu, Counting Sums of Two Squares: The Meissel-Lehmer Method, Mathematics of Computation 47:175 (July 1986), pp. 351-360. [Beware errors.]
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant

Cf. A000050, A001481, A064533, A227158, A275649, A275650.

a(n) = A180416(n) + ceiling(sqrt(10^n)). - Hiroaki Yamanouchi, Jul 14 2014

Offset changed from 0 to 1 by Robert G. Wilson v, Aug 29 2009
a(9) from Eric W. Weisstein, Aug 29 2009
a(10) from Donovan Johnson, Sep 16 2009
a(11)-a(12) from Ant King, May 02 2010
a(11)-a(12) corrected and a(13)-a(16) added by Hiroaki Yamanouchi, Jul 14 2014

A000205 Number of positive integers <= 2^n of form x^2 + 3 y^2.

Original entry on oeis.org

1, 1, 3, 4, 8, 14, 25, 45, 82, 151, 282, 531, 1003, 1907, 3645, 6993, 13456, 25978, 50248, 97446, 189291, 368338, 717804, 1400699, 2736534, 5352182, 10478044, 20531668, 40264582, 79022464, 155196838, 304997408, 599752463, 1180027022, 2322950591, 4575114295

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).

Seth A. Troisi, Table of n, a(n) for n = 0..45
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.
D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.
Index entries for sequences related to populations of quadratic forms

Cf. A000050, A003136.

(* This program is not suitable to compute more than a score of terms. *)
a[0] = a[1] = 1; a[n_] := a[n] = Module[{cnt, k, r, x, y}, For[cnt = a[n-1]; k = 2^(n-1)+1, k <= 2^n, k++, r = Reduce[x >= 0 && y >= 0 && k == x^2 + 3 y^2, {x, y}, Integers]; If[r =!= False, cnt++]]; cnt];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 23 2019 *)

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Extensions

A000074 Number of odd integers <= 2^n of form x^2 + y^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions

A000692 An approximation to population of x^2 + y^2 <= 2^n.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A164775 a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A000205 Number of positive integers <= 2^n of form x^2 + 3 y^2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica