cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A004144 Nonhypotenuse numbers (indices of positive squares that are not the sums of 2 distinct nonzero squares).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 107, 108, 112, 114, 118, 121, 124, 126, 127
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Also numbers with no prime factors of form 4*k+1.
m is a term iff A072438(m) = m.
Density 0. - Charles R Greathouse IV, Apr 16 2012
Closed under multiplication. Primitive elements are A045326, 2 and the primes of form 4*k+3. - Jean-Christophe Hervé, Nov 17 2013

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Complement of A009003.
Cf. A000290, A002145, A005089, A072437, A244659, A244662.
The subsequence of primes is A045326.

Programs

  • Haskell
    import Data.List (elemIndices)
a004144 n = a004144_list !! (n-1)
a004144_list = map (+ 1) $ elemIndices 0 a005089_list
-- Reinhard Zumkeller, Jan 07 2013
  • Mathematica
    fQ[n_] := If[n > 1, First@ Union@ Mod[ First@# & /@ FactorInteger@ n, 4] != 1, True]; Select[ Range@ 127, fQ]
A004144 = Select[Range[127],Length@Reduce[s^2 + t^2 == s # && s > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 09 2020 *)
  • PARI
    is(n)=n==1||vecmin(factor(n)[,1]%4)>1 \\ Charles R Greathouse IV, Apr 16 2012
  • PARI
    list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=1,lim, if(u[i]==0, listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

Formula

A005089(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013
The number of terms below x is ~ (A * x / sqrt(log(x))) * (1 + C/log(x) + O(1/log(x)^2)), where A = A244659 and C = A244662 (Shanks, 1975). - Amiram Eldar, Jan 29 2022

Extensions

More terms from Reinhard Zumkeller, Jun 17 2002
Name clarified by Evan M. Bailey, Sep 17 2019

A088539 Decimal expansion of (4K/Pi)^2 where K is the Landau-Ramanujan constant.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Nov 16 2003

Keywords

Examples

			0.9468064071800793342160944131097562332500695...

References

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

Links

Crossrefs

Cf. A002144, A064533, A243379, A243380. Square of A244659.

Programs

  • Mathematica
    digits = 104; 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*LandauRamanujanK/Pi)^2 // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013, updated Mar 14 2018 *)

Formula

Equals prod(1-1/p^2) where p runs through the primes p==1 mod 4
A088539 * A243379 = 8 / Pi^2. - Vaclav Kotesovec, Apr 30 2020
Equals 1/A175647. - Vaclav Kotesovec, May 05 2020

A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 04 2014

Keywords

Examples

			0.70475345170594788412255819759189881852...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Links

Crossrefs

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

Programs

  • Mathematica
    digits = 100; 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]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

Formula

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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