

A192787


Number of distinct solutions of 4/n = 1/a + 1/b + 1/c in positive integers satisfying 1 <= a <= b <= c.


15



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 ErdősStraus 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 ErdősStraus equation in unit fractions, arXiv:1107.1010 [math.NT], 20102015.
Maria Monks, Ameya Velingker, On the ErdősStraus conjecture: Properties of solutions to its underlying diophantine equation. See also.
Alan Swett, The ErdosStrauss Conjecture, 1999.
R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), pp. 193198.
Wikipedia, ErdősStraus conjecture.
Konstantine Zelator, An ancient Egyptian problem: the diophantine equation 4/n=1/x+1/y+1/z, n>or=2, arXiv:0912.2458 [math.GM], 2009.


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=t1/a; for(b=max(1\t1+1, a), 2\t1, c=1/(t11/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=t1/a; for(b=max(1\t1+1, a), 2\t1, c=1/(t11/b); if(denominator(c)==1&&c>=b, s++; print("4/", n, " = 1/", a, " + 1/", b, " + 1/", c)))); s \\ Robert Munafo, Feb 19 2013


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.
Cf. A337432 (solutions with minimal c).
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 at the suggestion of Allan C. Wechsler by Charles R Greathouse IV, Feb 19 2013
Examples and crossreferences added by M. F. Hasler, Feb 19 2013


STATUS

approved



