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A292624
Number of solutions to 4/p = 1/x + 1/y + 1/z in positive integers, where p is the n-th prime.
6
3, 12, 12, 36, 48, 24, 24, 60, 120, 42, 108, 54, 42, 78, 198, 78, 156, 66, 96, 234, 42, 216, 156, 60, 48, 96, 156, 144, 90, 78, 192, 186, 102, 210, 108, 180, 144, 138, 384, 156, 276, 102, 396, 36, 138, 246, 174, 342, 216, 120, 114, 630, 48, 300
OFFSET
1,1
COMMENTS
Corrected version of A192788.
REFERENCES
For references and links see A192787.
LINKS
Christian Elsholtz, Terence Tao, Counting the number of solutions to the Erdos-Straus equation on unit fractions, arXiv:1107.1010 [math.NT], 2011-2015.
Christian Elsholtz, Terence Tao, Counting the number of solutions to the Erdos-Straus equation on unit fractions, Journal of the Australian Mathematical Society, 94(1), 50-105, 2013. doi:10.1017/S1446788712000468.
FORMULA
a(n) = A292581(A000040(n)).
EXAMPLE
a(3) = 12 because 4/(3rd prime) = 4/5 can be expressed in the following 12 ways:
4/5 = 1/2 + 1/4 + 1/20
4/5 = 1/2 + 1/5 + 1/10
4/5 = 1/2 + 1/10 + 1/5
4/5 = 1/2 + 1/20 + 1/4
4/5 = 1/4 + 1/2 + 1/20
4/5 = 1/4 + 1/20 + 1/2
4/5 = 1/5 + 1/2 + 1/10
4/5 = 1/5 + 1/10 + 1/2
4/5 = 1/10 + 1/2 + 1/5
4/5 = 1/10 + 1/5 + 1/2
4/5 = 1/20 + 1/2 + 1/4
4/5 = 1/20 + 1/4 + 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[Prime[n]]]; Sow[an]]][[2, 1]] (* Jean-François Alcover, Dec 02 2018, adapted from PARI *)
PROG
(PARI)
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)
)
}
a292624(n) =
{
local(t, t1, s, a, b, c);
t = 4/prime(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(a292624(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Sep 20 2017
STATUS
approved