A226168 Numbers n such that 1/a + 1/b + 1/c + 1/a*b*c = m /(a+b+c) where a, b and c are the 3 distinct prime divisors of n, and m is a positive integer such that the equation has infinitely many solutions.

42, 70, 84, 126, 140, 168, 231, 252, 280, 294, 336, 350, 378, 490, 504, 560, 588, 672, 693, 700, 756, 882, 980, 1008, 1120, 1134, 1176, 1344, 1400, 1512, 1617, 1750, 1764, 1960, 2016, 2058, 2079, 2240, 2268, 2352, 2450, 2541, 2646, 2688, 2800, 3024, 3402, 3430

Offset

1,1

Comments

Subset of A033992.

The value m = 12 is probably unique. We find only 3 primitive values of n: 42 = 2*3*7, 70 = 2*5*7 and 231 = 3*7*11.

Links

Table of n, a(n) for n=1..48.

Peter Vandendriessche, Hojoo Lee, Problems in Elementary Number Theory (see problem H67, p. 40). [Via Wayback Machine]

Example

42 is in the sequence because the prime divisors of 42 are 2, 3 and 7 => 1/2 + 1/3 + 1/7 + 1/(2*3*7) = 12/(2+3+7) = 1.

Maple

with(numtheory): for n from 2 to 3500 do:x:=factorset(n): n1:=nops(x): if n1=3 then x1:=x[1]:x2:=x[2]:x3:=x[3]:s:=1/x1+ 1/x2+ 1/x3+1/(x1*x2*x3): for m from 1 to 500 do:if s=m/(x1+x2+x3) then printf ( "%d %d \n", n, m):else fi:od:fi:od:

Crossrefs

Cf. A033992.

Sequence in context: A255989 A192274 A291319 * A248430 A340384 A330893

Adjacent sequences: A226165 A226166 A226167 * A226169 A226170 A226171

Keyword

nonn

Author

Michel Lagneau, May 29 2013

Status

approved