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.

A281873 a(n+1) is the smallest number greater than a(n) such that Sum_{j=1..n+1} 1/a(j) <= 4, a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 200, 77706, 16532869712, 3230579689970657935732, 36802906522516375115639735990520502954652700
Offset: 1

Views

Author

Yuriy Sibirmovsky, Jan 31 2017

Keywords

Comments

The method for any number A is to find the largest harmonic number H(n) smaller than A, then use the greedy algorithm to expand the difference A - H(n).
A140335 is the same sequence for 3. The sequence for 5 consists of 99 terms, the largest of which has 142548 digits.

References

  • A. M. Gleason, R. E. Greenwood, and L. M. Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA, 1980, pages 398-399.

Links

Crossrefs

Cf. A140335, A306349.

Programs

  • Mathematica
    x0=4-Sum[1/k,{k,1,30}];
Nm=10;
j=0;
While[x0>0||j==Nm,a0=Ceiling[1/x0];
x0=x0-1/a0;
Print[a0];j++]
f[s_List, n_] := Block[{t = Total[1/s]}, Append[s, Max[ s[[-1]] +1, Ceiling[1/(n - t)]]]]; Nest[f[#, 4] &, {1}, 34] (* Robert G. Wilson v, Feb 05 2017 *)
  • Python
    from sympy import egyptian_fraction
print(egyptian_fraction((4, 1))) # Pontus von Brömssen, Feb 10 2019

Formula

Sum_{k=1..35} 1/a(k) = 4.

A306349 Number of terms in the greedy Egyptian fraction representation of n.

Original entry on oeis.org

1, 4, 13, 35, 99
Offset: 1

Views

Author

Pontus von Brömssen, Feb 09 2019

Keywords

Comments

a(n) >= A004080(n).
a(6) > 255 and the denominator of the 255th term in the representation of 6 has 1264021241 digits.

Examples

			a(3)=13 is the number of terms of A140335;
a(4)=35 is the number of terms of A281873.

Links

Crossrefs

Cf. A004080, A140335, A281873.

Programs

  • Python
    from sympy import egyptian_fraction
def A306349(n): return len(egyptian_fraction((n,1)))

A323725 a(n) is the last (and thus largest) denominator of an Egyptian fraction representing n, where each consecutive denominator is as small as possible.

Original entry on oeis.org

1, 6, 57960, 36802906522516375115639735990520502954652700
Offset: 1

Views

Author

AJ Tatum, Aug 31 2019

Keywords

Comments

Values grow extremely quickly, a(5) has 142548 decimal digits.
The denominators for n = 3 are given in A140335.
The denominators for n = 4 are given in A281873.
The number of terms in the representation of n is A306349(n).

Examples

			a(3) = 57960 because (1/1) + (1/2) + (1/3) + (1/4) + (1/5) + (1/6) + (1/7) + (1/8) + (1/9) + (1/10) + (1/15) + (1/230) + (1/57960) = 3 and the final and greatest denominator is 57960. (Sequence A140335 has the full list of denominators.)

Links

Crossrefs

A140335 and A281873 are the denominatorial sequences for 3 and 4, respectively.
Cf. A306349.

Programs

  • PARI
    a(n)={my(s=n,k=1); while(s>1/k, s-=1/k; k++); while(s!=0, k=ceil(1/s); s-=1/k); k} \\ Andrew Howroyd, Sep 01 2019
  • Python
    from sympy import egyptian_fraction
def A323725(n): return egyptian_fraction((n,1))[-1] # Pontus von Brömssen, Aug 03 2020
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.