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 A192787 Number of distinct solutions of 4/n = 1/a + 1/b + 1/c in positive integers satisfying 1 <= a <= b <= c. 16
 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-Straus 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 [math.NT], 2010-2015. Maria Monks, Ameya Velingker, On the Erdős-Straus conjecture: Properties of solutions to its underlying diophantine equation. See also. Alan Swett, The Erdos-Strauss Conjecture, 1999. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), pp. 193-198. Wikipedia, Erdős-Straus 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, show=0)=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++; show&&print("4/", n, " = 1/", a, " + 1/", b, " + 1/", c)))); s \\ variant with print(...) added by Robert Munafo, Feb 19 2013, both combined through option "show" by M. F. Hasler, Jul 02 2022 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 cross-references added by M. F. Hasler, Feb 19 2013 STATUS approved

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