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 A292581 Number of solutions to 4/n = 1/x + 1/y + 1/z in positive integers. 7
 0, 3, 12, 10, 12, 39, 36, 46, 30, 63, 48, 106, 24, 93, 216, 148, 24, 141, 60, 196, 162, 141, 120, 304, 60, 111, 162, 232, 42, 459, 108, 394, 174, 141, 372, 442, 54, 183, 420, 538, 42, 489, 78, 352, 540, 243, 198, 904, 102, 303, 294, 334, 78, 513 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Corrected version of A192786. The Erdos-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. After a(2) = 3, the values shown are all composite. [Jonathan Vos Post, Jul 17 2011] LINKS Hugo Pfoertner, Table of n, a(n) for n = 1..1000 Christian Elsholtz and Terence Tao, Counting the number of solutions to the Erdos-Straus equation on unit fractions, arXiv:1107.1010 [math.NT], 2011-2015. Allan Swett, The Erdos-Strauss Conjecture, 1999. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), pp. 193-198. EXAMPLE a(3)=12 because 4/3 can be expressed in 12 ways: 4/3 = 1/1 + 1/4 + 1/12 4/3 = 1/1 + 1/6 + 1/6 4/3 = 1/1 + 1/12 + 1/4 4/3 = 1/2 + 1/2 + 1/3 4/3 = 1/2 + 1/3 + 1/2 4/3 = 1/3 + 1/2 + 1/2 4/3 = 1/4 + 1/1 + 1/12 4/3 = 1/4 + 1/12 + 1/1 4/3 = 1/6 + 1/1 + 1/6 4/3 = 1/6 + 1/6 + 1/1 4/3 = 1/12 + 1/1 + 1/4 4/3 = 1/12 + 1/4 + 1/1 a(4) = 10 because 4/4 = 1 can be expressed in 10 ways: 4/4= 1/2 + 1/3 + 1/6 4/4= 1/2 + 1/4 + 1/4 4/4= 1/2 + 1/6 + 1/3 4/4= 1/3 + 1/2 + 1/6 4/4= 1/3 + 1/3 + 1/3 4/4= 1/3 + 1/6 + 1/2 4/4= 1/4 + 1/2 + 1/4 4/4= 1/4 + 1/4 + 1/2 4/4= 1/6 + 1/2 + 1/3 4/4= 1/6 + 1/3 + 1/2 MATHEMATICA checkmult[a_, b_, c_] := If[Denominator[c] == 1, If[a == b && a == c && b == c, Return[1], If[a != b && a != c && b != c, Return[6], Return[3]]], Return[0]]; a292581[n_] := Module[{t, t1, s, a, b, c, q = Quotient}, t = 4/n; s = 0; For[a = q[1, t]+1, a <= q[3, t], a++, t1 = t - 1/a; For[b = Max[q[1, t1] + 1, a], b <= q[2, t1], b++, c = 1/(t1 - 1/b); s += checkmult[a, b, c]]]; Return[s]]; Reap[For[n=1, n <= 54, n++, Print[n, " ", an = a292581[n]]; Sow[an]]][[2, 1]] (* Jean-François Alcover, Dec 02 2018, adapted from PARI *) PROG (PARI) \\ modified version of code by Charles R Greathouse IV in A192786 checkmult (a, b, c) = { if(denominator(c)==1, if(a==b && a==c && b==c, return(1), if(a!=b && a!=c && b!=c, return(6), return(3) ) ), return(0) ) } a292581(n) = { local(t, t1, s, a, b, c); t = 4/n; s = 0; 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); s+=checkmult(a, b, c); ) ); return(s); } for (n=1, 54, print1(a292581(n), ", ")) CROSSREFS For more references and links see A192787. Cf. A073101, A192786, A292624, A337432. Sequence in context: A215842 A018876 A038230 * A207852 A182455 A110345 Adjacent sequences: A292578 A292579 A292580 * A292582 A292583 A292584 KEYWORD nonn AUTHOR Hugo Pfoertner, Sep 20 2017 STATUS approved

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Last modified June 17 15:28 EDT 2024. Contains 373456 sequences. (Running on oeis4.)