login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192787 Number of distinct solutions of 4/n = 1/a + 1/b + 1/c in positive integers satisfying 1 <= a <= b <= c. 14
0, 1, 3, 3, 2, 8, 7, 10, 6, 12, 9, 21, 4, 17, 39, 28, 4, 26, 11, 36, 29, 25, 21, 57, 10, 20, 29, 42, 7, 81, 19, 70, 31, 25, 65, 79, 9, 32, 73, 96, 7, 86, 14, 62, 93, 42, 34, 160, 18, 53, 52, 59, 13, 89, 98, 136, 41, 33, 27, 196, 11, 37, 155, 128, 49, 103, 17, 73, 55, 185, 40, 211, 7, 32, 129, 80, 97, 160, 37, 292 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Erdős-Strauss conjecture is that a(n) > 0 for n > 1. Swett verified the conjecture for n < 10^14.

Vaughan shows that the number of n < x with a(n) = 0 is at most x exp(-c * (log x)^(2/3)) for some c > 0.

See A073101 for the 4/n conjecture due to Erdős and Straus.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Christian Elsholtz, Terence Tao, Counting the number of solutions to the Erdős-Straus equation in unit fractions, arXiv:1107.1010

Maria Monks, Ameya Velingker, On the Erdős-Straus conjecture: Properties of solutions to its underlying diophtanine equation

Alan Swett, The Erdos-Strauss Conjecture, 1999

R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), pp. 193-198.

Konstantine Zelator, An ancient Egyptian problem: the diophantine equation 4/n=1/x+1/y+1/z, n>or=2, arXiv:0912.2458

EXAMPLE

a(1) = 0, since 4/1 = 4 cannot be expressed as the sum of three reciprocals.

a(2) = 1 because 4/2 = 1/1 + 1/2 + 1/2, and there are no other solutions.

a(3) = 3 since 4/3 = 1 + 1/4 + 1/12 = 1 + 1/6 + 1/6 = 1/2 + 1/2 + 1/3.

a(4) = 3 = A002966(3).

MAPLE

A192787 := proc(n) local t, a, b, t1, count; t:= 4/n; count:= 0; for a from floor(1/t)+1 to floor(3/t) do t1:= t - 1/a; for b from max(a, floor(1/t1)+1) to floor(2/t1) do if type( 1/(t1 - 1/b), integer) then count:= count+1; end if end do end do; count; end proc; # Robert Israel, Feb 19 2013

MATHEMATICA

f[n_] := Length@ Solve[ 4/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Array[f, 70] (* Allan C. Wechsler and Robert G. Wilson v, Jul 19 2013 *)

PROG

(PARI) a(n)=my(t=4/n, t1, s, c); for(a=1\t+1, 3\t, t1=t-1/a; for(b=max(1\t1+1, a), 2\t1, c=1/(t1-1/b); if(denominator(c)==1&&c>=b, s++))); s

(PARI) a(n)=my(t=4/n, t1, s, c); for(a=1\t+1, 3\t, t1=t-1/a; for(b=max(1\t1+1, a), 2\t1, c=1/(t1-1/b); if(denominator(c)==1&&c>=b, s++; print("4/", n, " = 1/", a, " + 1/", b, " + 1/", c)))); s

CROSSREFS

A292581 counts the solutions with multiplicity. A073101 counts solutions with a, b, and c distinct.

Cf. A292581, A292624, A192789, A075245, A075246, A075247, A075248.

Cf. A004194, A226641, A226642, A226644, A226645, A226646.

Sequence in context: A238278 A200770 A265965 * A268724 A248569 A214101

Adjacent sequences:  A192784 A192785 A192786 * A192788 A192789 A192790

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Jul 10 2011

EXTENSIONS

Corrected on the suggestion of Allan C. Wechsler by Charles R Greathouse IV, Feb 19 2013

Examples and cross-references added by M. F. Hasler, Feb 19 2013

Printing version of PARI code by Robert Munafo, Feb 19 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)