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 1-4 of 4 results.

A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

Original entry on oeis.org

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

Views

Author

Labos Elemer, Oct 04 2001

Keywords

Comments

The Engel expansion of this constant is the sequence of primes. - Jonathan Vos Post, May 04 2005
Let S be the operator over the space omega of infinite sequences of numbers, defined to be the Engel expansion of the sum of reciprocals of primorials of a sequence p of numbers; than the eigenvalue-equation S p = p is satisfied by the sequence of prime numbers. - Ralf Steiner, Dec 31 2016
This constant is irrational (Griffiths, 2015). - Amiram Eldar, Oct 27 2020

Examples

			0.705230171791800965147431682888248513743577639109154328192267913813919...

References

  • Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Links

Crossrefs

Cf. A002110, A054543, A000027, A053977, A006784, A028259, A165509 (continued fraction).
Cf. A249270, A340469.

Programs

  • Mathematica
    RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (* Robert G. Wilson v, Aug 05 2005 *)
RealDigits[Total[1/#&/@FoldList[Times,Prime[Range[100]]]],10,120][[1]] (* Harvey P. Dale, Aug 27 2019 *)
  • PARI
    default(realprecision, 20080); p=1; s=x=0; for (k=1, 10^9, p*=prime(k); s+=1.0/p; if (s==x, break); x=s ); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b064648.txt", n, " ", d)) \\ Harry J. Smith, Sep 21 2009
  • Sage
    @CachedFunction
def pv(n):
    a = 1
    b = 0
    for i in (1..n):
        a *= nth_prime(i)
        b += 1/a
    return b
N(pv(100),digits=108) # From Maple code Jani Melik, Jul 22 2015

Formula

(1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - Jonathan Sondow, Aug 04 2014
Equals Sum_{n>=1} 1/A002110(n). - Amiram Eldar, Oct 27 2020

A220398 A modified Engel expansion of the golden ratio (1/2)*(1 + sqrt(5)) (A001622).

Original entry on oeis.org

1, 2, 5, 8, 3, 4, 4, 6, 2, 162, 322, 2, 51842, 103682, 2, 5374978562, 10749957122, 2, 57780789062419261442, 115561578124838522882, 2, 6677239169351578707225356193679818792962, 13354478338703157414450712387359637585922, 2
Offset: 1

Views

Author

Peter Bala, Dec 13 2012

Keywords

Comments

See A220393 for a definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Links

Crossrefs

Cf. A001622, A028259, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220397.

Formula

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = 1/2*(1 + sqrt(5)) - 1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).
Recurrence equations: For n >= 3, a(3*n) = 2. For n >= 4 we have a(3*n+2) = 2*a(3*n+1) - 2 and a(3*n+1) = 2*(a(3*n-2) - 1)^2.
Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1 + 1/2 + 1/(2*5) + 1/(2*5*8) + 1/(2*5*8*3) + 1/(2*5*8*3*4) + .... For n >= 2, the error made in truncating this series to n terms is less than the n-th term.

A059176 Engel expansion of sqrt(5) = 2.23606...

Original entry on oeis.org

1, 1, 5, 6, 13, 16, 16, 38, 48, 58, 104, 177, 263, 332, 389, 4102, 4575, 5081, 9962, 18316, 86613, 233239, 342534, 964372, 1452850, 7037119, 7339713, 8270361, 12855437, 15900982, 19211148, 1365302354, 1565752087, 1731612283
Offset: 1

Views

Author

Mitch Harris

Keywords

Comments

Cf. A006784 for definition of Engel expansion.

References

  • F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

Links

Crossrefs

Cf. A002163.
Essentially the same as A028259.

Programs

  • Mathematica
    EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Sqrt[5], 7!], 50] (* modified by G. C. Greubel, Dec 26 2016 *)

A278765 Engel expansion of natural logarithm of golden ratio.

Original entry on oeis.org

3, 3, 4, 4, 4, 6, 15, 48, 118, 147, 671, 1026, 3075, 44641, 48364, 1868380, 75080506, 96848501, 911582093, 2511879981, 8700005050, 15888441652, 108526838262, 446779835336, 632466801279, 1084794852728, 1184722346307, 1657692322844, 12376968750845, 17341469111712, 27996895637798, 38935285631573
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 28 2016

Keywords

Examples

			log(phi)  = 0.4812118250596... = 1/3 + 1/(3*3) + 1/(3*3*4) + 1/(3*3*4*4) + 1/(3*3*4*4*4) + 1/(3*3*4*4*4*6) + ...

Links

Crossrefs

Cf. A006784 (for definition of Engel expansion).
Cf. A001622, A002390, A028259, A059195.

Programs

  • Mathematica
    EngelExp[A_, n_]:=Join[Array[1&, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]], Expand[ #[[1]]#[[2]]-1]}&, {Ceiling[1/(A-Floor[A])], A-Floor[A]}, n-1]]; EngelExp[N[Log[GoldenRatio], 7! ], 40]
Showing 1-4 of 4 results.

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