A004194 Number of partitions of 1/n into 3 reciprocals of positive integers.

3, 10, 21, 28, 36, 57, 42, 70, 79, 96, 62, 160, 59, 136, 196, 128, 73, 211, 80, 292, 245, 157, 93, 366, 156, 174, 230, 340, 106, 497, 90, 269, 322, 211, 453, 538, 85, 216, 378, 604, 121, 623, 104, 473, 648, 204, 135, 706, 227, 437, 387, 467, 125, 601, 561, 783, 385

Offset: 1

Author

Scott Aaronson (philomath(AT)voicenet.com)

Keywords

• nonn

Comments

Number of ways to express 1/n as Egyptian fractions in just three terms; i.e., 1/n = 1/x + 1/y + 1/z satisfying 1<=x<=y<=z.

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

Links

• Justus Springer, Table of n, a(n) for n = 1..5000 (terms 1..100 from Robert G. Wilson)
• K. S. Brown, Unit Fraction Partitions
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A227610, A226641, A226642, A192787, A226644, A226645, A226646.

Programs

Mathematica

a[n_] := Length@ Solve[ 1/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Array[a, 70] (* Allan C. Wechsler and Robert G. Wilson v, Aug 17 2013 *)

PARI

a(n)=my(t=1/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 \\ Charles R Greathouse IV, Jun 12 2013

Extensions

More terms from David W. Wilson, Aug 15 1996