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

1,1

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Project Euler, Problem 221: Alexandrian integers

Example

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

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

Prog

(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))) }

Crossrefs

Cf. A147807, A147808, A147809, A147810.

Sequence in context: A153786 A256833 A164016 * A046763 A199905 A360757

Adjacent sequences: A147808 A147809 A147810 * A147812 A147813 A147814

Keyword

nonn

Author

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

Status

approved