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.

A257840 y-value of the lexicographically first integer solution (x,y,z) of 4/n = 1/x + 1/y + 1/z with 0 < x < y < z, or 0 if there is no such solution. Corresponding x and z values are in A257839 and A257841.

Original entry on oeis.org

0, 0, 4, 3, 4, 7, 15, 7, 10, 16, 34, 13, 18, 29, 61, 21, 30, 46, 96, 31, 43, 67, 139, 43, 60, 92, 190, 57, 78, 121, 249, 73, 100, 154, 316, 91, 124, 191, 391, 111, 154, 232, 474, 133, 181, 277, 565, 157, 99, 326, 664, 183, 248, 379, 771, 211, 286, 436, 886, 241, 326, 497, 1009, 273, 370, 562, 1140, 307, 415, 631, 1279, 343, 210, 704, 1426, 381, 514, 781, 1581, 421
Offset: 1

Views

Author

M. F. Hasler, May 16 2015

Keywords

Comments

See A073101 for more details.
This differs from A075246 starting with a(89)=690 vs A075246(89)=306, corresponding to the representations 4/89 = 1/23 + 1/690 + 1/61410 = 1/24 + 1/306 + 1/108936.

Links

Crossrefs

Cf. A073101, A075245, A075246, A075247.

Programs

  • PARI
    apply( {A257840(n, t)=for(x=n\4+1, 3*n\4, for(y=max(1\t=4/n-1/x, x)+1, ceil(2/t)-1, numerator(t-1/y)==1 && return(y)))}, [1..99]) \\ improved by M. F. Hasler, Jul 03 2022

A257839 Smallest possible x such that 4/n = 1/x + 1/y + 1/z with 0 < x < y < z all integers, or 0 if there is no such solution. Corresponding y and z values are in A257840 and A257841.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 14, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 20, 19, 19, 20, 20, 20, 20, 21
Offset: 1

Views

Author

M. F. Hasler, May 16 2015

Keywords

Comments

Otherwise said, x-value of the lexicographically first solution (x,y,z) to the given equation.
See A073101 for more details about these sequences related to the Erdős-Straus conjecture.
This differs from A075245 starting with a(89)=23 vs A075245(89)=24, respective solutions being 1/23 + 1/690 + 1/61410 = 1/24 + 1/306 + 1/108936 = 4/89.

Links

Crossrefs

Cf. A073101, A075245, A075246, A075247.
Cf. A257840, A257841.

Programs

  • PARI
    apply( {A257839(n, t)=for(x=n\4+1, 3*n\4, for(y=max(1\t=4/n-1/x, x)+1, ceil(2/t)-1, numerator(t-1/y)==1 && return(x)))}, [1..99]) \\ improved by M. F. Hasler, Jul 03 2022

Formula

Conjecture: a(n) = floor(n/4) + d with d = 1 for all n > 2 except some n = 24k + 1 (k = 2, 3, 7, 8, 10, 13, 15, 17, 18, 23, 25, 28, 30, 32, 33, 37, 40, 43, ...) where d = 2. - M. F. Hasler, Jul 03 2022

A073101 Number of integer solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 5, 6, 4, 9, 7, 15, 4, 14, 33, 22, 4, 21, 9, 30, 25, 22, 19, 45, 10, 17, 25, 36, 7, 72, 17, 62, 27, 22, 59, 69, 9, 29, 67, 84, 7, 77, 12, 56, 87, 39, 32, 142, 16, 48, 46, 53, 13, 82, 92, 124, 37, 30, 25, 178, 11, 34, 147, 118, 49, 94, 15, 67, 51, 176, 38, 191, 7
Offset: 1

Views

Author

Robert G. Wilson v, Aug 18 2002

Keywords

Comments

In 1948 Erdős and Straus conjectured that for any positive integer n >= 2 the equation 4/n = 1/x + 1/y + 1/z has a solution with positive integers x, y and z (without the additional requirement 0 < x < y < z). All of the solutions can be printed by removing the comment symbols from the Mathematica program. For the solution (x,y,z) having the largest z value, see (A075245, A075246, A075247). See A075248 for Sierpiński's conjecture for 5/n.
See (A257839, A257840, A257841) for the lexicographically smallest solutions, and A257843 for the differences between these and those with largest z-value. - M. F. Hasler, May 16 2015

Examples

			a(5)=2 because there are two solutions: 4/5 = 1/2 + 1/4 + 1/20 and 4/5 = 1/2 + 1/5 + 1/10.

Links

Crossrefs

Cf. A075245, A075246, A075247, A075248.
Cf. A192787 (# distinct solutions with x <= y <= z).

Programs

  • Haskell
    import Data.Ratio ((%), numerator, denominator)
a073101 n = length [(x,y) |
   x <- [n `div` 4 + 1 .. 3 * n `div` 4],   let y' = recip $ 4%n - 1%x,
   y <- [floor y' + 1 .. floor (2*y') + 1], let z' = recip $ 4%n - 1%x - 1%y,
   denominator z' == 1 && numerator z' > y && y > x]
-- Reinhard Zumkeller, Jan 03 2011
  • Maple
    A:= proc(n)
   local x,t, p,q,ds,zs,ys,js, tot,j;
tot:= 0;
for x from 1+floor(n/4) to ceil(3*n/4)-1 do
    t:= 4/n - 1/x;
    p:= numer(t);
    q:= denom(t);
    ds:= convert(select(d -> (d < q) and d + q mod p = 0,
          numtheory:-divisors(q^2)),list);
    ys:= map(d -> (d+q)/p, ds);
    zs:= map(d -> (q^2/d+q)/p, ds);
    js:= select(j -> ys[j] > x,[$1..nops(ds)]);
    tot:= tot + nops(js);
od;
tot;
end proc:
seq(A(n),n=2..100); # Robert Israel, Aug 22 2014
  • Mathematica
    (* download Egypt.m from D. Eppstein's site and put it into MyOwn directory underneath Mathematica\AddOns\StandardPackages *) Needs["MyOwn`Egypt`"]; Table[ Length[ EgyptianFraction[4/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Disallow, OutputFormat -> Plain]], {n, 5, 80}]
m = 4; For[lst = {}; n = 2, n <= 100, n++, cnt = 0; xr = n/m; If[IntegerQ[xr], xMin = xr + 1, xMin = Ceiling[xr]]; If[IntegerQ[3xr], xMax = 3xr - 1, xMax = Floor[3xr]]; For[x = xMin, x <= xMax, x++, yr = 1/(m/n - 1/x); If[IntegerQ[yr], yMin = yr + 1, yMin = Ceiling[yr]]; If[IntegerQ[2yr], yMax = 2yr + 1, yMax = Ceiling[2yr]]; For[y = yMin, y <= yMax, y++, zr = 1/(m/n - 1/x - 1/y); If[y > x && zr > y && IntegerQ[zr], z = zr; cnt++; (*Print[n, " ", x, " ", y, " ", z]*)]]]; AppendTo[lst, cnt]]; lst
f[n_] := Length@ Solve[4/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 72, 2] (* Robert G. Wilson v, Jul 17 2013 *)
  • PARI
    A073101(n)=sum(c=n\4+1,n*3\4,sum(b=c+1,ceil(2/(t=4/n-1/c))-1,numerator(t-1/b)==1)) \\ M. F. Hasler, May 15 2015

Extensions

Edited by T. D. Noe, Sep 10 2002
Extended to offset 1 with a(1) = 0 by M. F. Hasler, May 16 2015

A257843 Numbers n for which the lexicographically first integer solution (x,y,z) of 4/n = 1/x + 1/y + 1/z with 0 < x < y < z, is different from the solution having the largest z-value.

Original entry on oeis.org

89, 113, 161, 233, 281, 329, 353, 401, 409, 449, 473, 521, 593, 641, 689, 713, 761, 769, 809, 929, 953, 1049, 1073, 1121, 1129, 1169, 1193, 1241, 1249, 1313, 1321, 1361, 1369, 1409, 1433, 1481, 1513, 1529, 1553, 1561, 1601, 1609, 1649, 1673, 1721, 1769
Offset: 1

Views

Author

M. F. Hasler, May 16 2015

Keywords

Comments

Related to the Erdős-Straus conjecture, see A073101 for more details.
This lists indices for which (A075245, A075246, A075247) differ from (A257839, A257840, A257841).

Examples

			For n=89, 4/89 = 1/23 + 1/690 + 1/61410 = 1/24 + 1/306 + 1/108936 are the representations with the largest resp. smallest unit fraction.

Links

Crossrefs

Cf. A073101, A075245, A075246, A075247, A257839, A257840, A257841.

Programs

  • PARI
    is(n,s=0)=for(c=n\4+1,n*3\4,for(b=c+1,ceil(2/(t=4/n-1/c))-1,numerator(t-1/b)==1||next;!s&&(s=t-1/b)&&next(2);t-1/b

Extensions

More terms from Manfred Scheucher, May 24 2015
Showing 1-4 of 4 results.

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