A004018 -id:A004018 - cp's OEIS frontend

A331071 a(n) = Sum_{k <= n} r_2(k)^2*d(k+1), where r_2 = A004018, d = A000005.

1, 33, 65, 65, 97, 353, 353, 353, 401, 465, 593, 593, 593, 849, 849, 849, 881, 1265, 1297, 1297, 1553, 1553, 1553, 1553, 1553, 2129, 2385, 2385, 2385, 2897, 2897, 2897, 2961, 2961, 3217, 3217, 3249, 3505, 3505, 3505, 3633, 4145, 4145, 4145, 4145, 4401, 4401, 4401, 4401, 4497, 5073, 5073, 5201, 5713, 5713

Offset: 0

N. J. A. Sloane, Jan 10 2020

nonn

Partial sums of A330574.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 163.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Karl-Heinz Indlekofer, Eine asymptotische Formel in der Zahlentheorie, (German) Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).

Cf. A000005, A004018, A330574.

Accumulate @ Table[SquaresR[2, n]^2 * DivisorSigma[0, n+1], {n, 0, 50}] (* Amiram Eldar, Mar 05 2020 *)

a(n) ~ c * n * log(n)^2, where c is a constant. - Amiram Eldar, Mar 05 2020

A331134 a(n) = Sum_{primes p <= n} r_2(p)/4, where r_2(n) = A004018(n).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 21

Offset: 1

N. J. A. Sloane, Jan 11 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284).

Cf. A004018, A331135.

Accumulate[Table[If[PrimeQ[n], SquaresR[2, n], 0], {n, 1, 100}]/4] (* Amiram Eldar, Apr 23 2024 *)

A124118 Decimal expansion of Sum_{i>=0} A004018(i)/2^i.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 25 2006

			4.532372014258974100827957178...

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 6 ed., 2008, section 17.10, p. 340.

D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518. LBNL-53854.
Simon Plouffe, Constants derived from sums of A004018 [broken link?].

Cf. A004018.

Clear[s]; s[n_] := s[n] = RealDigits[ Sum[ SquaresR[2, k]/2^k, {k, 0, n}], 10, 29] // First; s[n=100]; While[s[n] != s[n-100], n = n+100]; s[n] (* Jean-François Alcover, Feb 13 2013 *)
RealDigits[1 + 4*Sum[(-1)^n/(2^(2*n + 1) - 1), {n, 0, 200}], 10, 100][[1]] (* Amiram Eldar, Jun 22 2020 *)

Sum_{i>=0} A004018(i)/2^i.
Bailey et al. point out the approximation Pi*(1+2*exp(-Pi^2/log(2))^2)/log(2), correct up to 23 decimal places. - Jean-François Alcover, Jun 27 2015
Equals 1 + 4 * Sum_{k>=0} (-1)^k/(2^(2*k+1) - 1). - Amiram Eldar, Jun 22 2020

A326311 Least numbers k such that A004018(k) is nondecreasing.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 13, 17, 20, 25, 50, 65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 325, 425, 650, 725, 845, 850, 925, 1025, 1105, 1625, 1885, 2125, 2210, 2405, 2465, 2665, 3145, 3250, 3445, 3485, 3625, 3770, 3965, 4225, 5525

Offset: 1

Hugo Pfoertner, Sep 11 2019

nonn

Least squared radius of a circle around a grid point of the square lattice such that the number of grid points on this circle is not smaller than the number of grid points on any circle around a grid point of the square lattice with smaller radius. a(1) = 0 by convention.

Hugo Pfoertner, Table of n, a(n) for n = 1..440

Cf. A004018, A071383, A071385, A326421.

using Nemo
function A326311List(len)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(2, len, x)
    L = [coeff(e, j) for j in 0:len - 1]
    m = ZZ(0)
    [n - 1 for (n, l) in enumerate(L) if l == (m = max(m, l))]
end
A326311List(1000) |> println # Peter Luschny, Sep 12 2019

r2=0;for(k=0,6000,my(a004018 = if( k<1, k==0, 4 * sumdiv( k, d, (d%4==1) - (d%4==3))));if(a004018>=r2,r2=a004018;print1(k,", ")))

A330574 a(n) = r_2(n)^2*d(n+1), where r_2 = A004018, d = A000005.

Original entry on oeis.org

1, 32, 32, 0, 32, 256, 0, 0, 48, 64, 128, 0, 0, 256, 0, 0, 32, 384, 32, 0, 256, 0, 0, 0, 0, 576, 256, 0, 0, 512, 0, 0, 64, 0, 256, 0, 32, 256, 0, 0, 128, 512, 0, 0, 0, 256, 0, 0, 0, 96, 576, 0, 128, 512, 0, 0, 0, 0, 128, 0, 0, 256, 0, 0, 64, 2048, 0, 0, 256, 0, 0, 0, 32, 256, 384, 0, 0, 0, 0, 0, 320, 64, 128, 0, 0

Offset: 0

N. J. A. Sloane, Jan 10 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Karl-Heinz Indlekofer, Eine asymptotische Formel in der Zahlentheorie, (German) Arch. Math. (Basel) 23 (1972), 619-624. MR0318080 (47 #6629).

Cf. A000005, A004018, A331071.

Table[SquaresR[2, n]^2 * DivisorSigma[0, n+1], {n, 0, 100}] (* Amiram Eldar, Mar 05 2020 *)

Offset corrected by Amiram Eldar, Mar 05 2020

A331135 a(n) = Sum_{primes p <= n} r_2(p-1)/4, where r_2(n) = A004018(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

N. J. A. Sloane, Jan 11 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284).

Cf. A004018, A331134.

Accumulate[Table[If[PrimeQ[n], SquaresR[2, n-1], 0], {n, 1, 100}]/4] (* Amiram Eldar, Apr 23 2024 *)

A141666 A symmetrical triangle of coefficients based on A004018 (or number of ways of writing n as a sum of 2 squares): t(n,m) = r2(n-m+1)*r2(m+1).

Original entry on oeis.org

1, 4, 4, 4, 16, 4, 0, 16, 16, 0, 4, 0, 16, 0, 4, 8, 16, 0, 0, 16, 8, 0, 32, 16, 0, 16, 32, 0, 0, 0, 32, 0, 0, 32, 0, 0, 4, 0, 0, 0, 16, 0, 0, 0, 4, 4, 16, 0, 0, 32, 32, 0, 0, 16, 4, 8, 16, 16, 0, 0, 64, 0, 0, 16, 16, 8

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 05 2008

Row sums are {1, 8, 24, 32, 24, 48, 96, 64, 24, 104, 144}.

			Triangle begins
  {1},
  {4,  4},
  {4, 16,  4},
  {0, 16, 16,  0},
  {4,  0, 16,  0,  4},
  {8, 16,  0,  0, 16,  8},
  {0, 32, 16,  0, 16, 32,  0},
  {0,  0, 32,  0,  0, 32,  0,  0},
  {4,  0,  0,  0, 16,  0,  0,  0,  4},
  {4, 16,  0,  0, 32, 32,  0,  0, 16,  4},
  {8, 16, 16,  0,  0, 64,  0,  0, 16, 16,  8}

G. E. Andrews, Number Theory, 1971, Dover Publications New York, p. 44, p. 201-207.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Sum of Squares Function

Cf. A004018.

Clear[a]; a = CoefficientList[Series[1 + 4*Sum[(-1)^(1 + n)/(-1 + x^(1 - 2*n)), {n, 100}], {x, 0, 100}], x]; Table[Table[a[[n - m + 1]]*a[[m + 1]], {m, 0, n}], {n, 0, 10}]//Flatten

t(n,m) = r2(n-m+1)*r2(m+1).

A331136 a(n) = Sum_{primes p < n} r_2(n-p)/4, where r_2(n) = A004018(n).

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 4, 3, 2, 3, 3, 6, 4, 2, 7, 5, 3, 6, 5, 7, 7, 7, 7, 7, 5, 3, 10, 10, 6, 8, 9, 9, 9, 7, 4, 13, 9, 6, 15, 7, 7, 14, 11, 10, 11, 9, 12, 16, 9, 7, 12, 13, 11, 15, 14, 13, 17, 12, 7, 16, 11, 8, 23, 12, 9, 17, 14, 16, 18, 15, 15, 21, 12, 10, 17, 19, 16, 20, 16, 11, 22, 15, 14, 27, 14, 11, 29, 20, 12, 18

Offset: 1

N. J. A. Sloane, Jan 11 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284).

Cf. A004018, A331134.

a[n_] := Sum[SquaresR[2, n - p]/4, {p, Select[Range[n - 1], PrimeQ]}]; Array[a, 100] (* Amiram Eldar, Apr 23 2024 *)

A333168 a(n) = Sum_{k=0..n} r_2(k^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

Original entry on oeis.org

4, 8, 16, 24, 32, 40, 48, 60, 76, 84, 92, 100, 116, 132, 140, 148, 156, 172, 196, 204, 212, 228, 244, 260, 268, 276, 284, 300, 316, 324, 340, 356, 380, 396, 412, 420, 428, 444, 468, 476, 484, 496, 512, 536, 552, 560, 576, 608, 624, 632, 648, 656, 672, 688, 696

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = r_2(0^2 + 1) = r_2(1) = A004018(1) = 4.
a(1) = r_2(0^2 + 1) + r_2(1^1 + 1) = r_2(1) + r_2(2) = A004018(1) + A004018(2) = 4 + 4 = 8.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

Amiram Eldar, Table of n, a(n) for n = 0..10000
E. J. Scourfield, The divisors of a quadratic polynomial, Glasgow Mathematical Journal, Vol. 5, No. 1 (1961), pp. 8-20.

Partial sums of A333167.

Cf. A002522, A004018.

Accumulate @ Table[SquaresR[2, k^2 + 1], {k, 0, 100}]

a(n) ~ (8/Pi) * n * log(n).

A046106 G.f.: g.f. for A001411 / g.f. for A004018.

Original entry on oeis.org

1, 0, 8, 4, 48, 68, 284, 684, 1816, 5608, 12684, 42068, 92916, 304100, 688988, 2170020, 5088784, 15436172, 37281880, 109786204, 271062388, 781016892, 1958863988, 5555714820, 14090644980, 39503105472, 101000072900, 280693435596

Offset: 0

N. J. A. Sloane, based on a suggestion of Maurice Craig (Maurice.D.Craig(AT)BHPBilliton.com), May 11 2003

nonn

			1 + 4*q + 12*q^2 + 36*q^3 + 100*q^4 + 284*q^5 + 780*q^6 + 2172*q^7 + 5916*q^8 + 16268*q^9 + ... / 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + ... = 1 + 8*q^2 + 4*q^3 + 48*q^4 + 68*q^5 + 284*q^6 + 684*q^7 + 1816*q^8 + 5608*q^9 + ...

Cf. A001411, A004018.

cp's OEIS Frontend

A331071 a(n) = Sum_{k <= n} r_2(k)^2*d(k+1), where r_2 = A004018, d = A000005.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A331134 a(n) = Sum_{primes p <= n} r_2(p)/4, where r_2(n) = A004018(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A124118 Decimal expansion of Sum_{i>=0} A004018(i)/2^i.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A326311 Least numbers k such that A004018(k) is nondecreasing.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Julia

PARI

A330574 a(n) = r_2(n)^2*d(n+1), where r_2 = A004018, d = A000005.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A331135 a(n) = Sum_{primes p <= n} r_2(p-1)/4, where r_2(n) = A004018(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A141666 A symmetrical triangle of coefficients based on A004018 (or number of ways of writing n as a sum of 2 squares): t(n,m) = r2(n-m+1)*r2(m+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A331136 a(n) = Sum_{primes p < n} r_2(n-p)/4, where r_2(n) = A004018(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A333168 a(n) = Sum_{k=0..n} r_2(k^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

Views