A147811 Alexandrian integers: numbers of the form n=pqr such that 1/n = 1/p - 1/q - 1/r for some integers p,q,r.

6, 42, 120, 156, 420, 630, 930, 1428, 1806, 2016, 2184, 3192, 4950, 5256, 8190, 8364, 8970, 10296, 10998, 12210, 17556, 19110, 21114, 23994, 24492, 28050, 32640, 33306, 34362, 37506, 39270, 44310, 52326, 57684, 57840, 70686, 74256, 79800, 83076

Offset: 1

Author

M. F. Hasler and Alexis Olson (AlexisOlson(AT)gmail.com), Dec 13 2008

Keywords

• nonn

Comments

The numbers are of the form p(p+d)(p+(p^2+1)/d), where d runs over divisors of p^2+1 and p runs over all positive integers. See also A147807..A147810. - M. F. Hasler, Jan 07 2009

Examples

			630 is an Alexandrian integer since 630 = 5(-7)(-18) and 1/630 = 1/5 - 1/7 - 1/18.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Project Euler, Problem 221: Alexandrian integers

Crossrefs

Cf. A147807, A147808, A147809, A147810.

Programs

Maple

N:= 10^5: # to get all terms <= N
A:= select(`<=`,{seq(seq(p*(p+d)*(p+(p^2+1)/d), d=numtheory:-divisors(p^2+1)),p=1..floor(N^(1/3)))},N):
sort(convert(A,list)); # Robert Israel, Dec 16 2018

PARI

is_A147811(n) = { my(d=divisors(n), c=#d+1); n<42 && return(n==6); for( i=2, c-3, d[i+1]^2>d[c-i] && return; d[c-i]%d[i]==1 | next; for( j=i+1, c-i,d[j]^2>d[c-i] && next(2); d[c-i]\d[j]*(d[j]-d[i]) == d[j]*d[i]+1 && return(1))) }