OFFSET
1,1
COMMENTS
Results for p = q are given in A024702, which is complementary.
All integer results when viewed in the triangle occur in loosely diagonal, interrupted "bands" roughly (or exactly) parallel to main diagonal, such that q - p = 24m, where m = 1 for the first band closest to the main diagonal, m = 2 for the second band, m = 3 for the third band, etc. The main diagonal p = q can be considered as fitting in this pattern where m = 0.
A general "rule" can be stated: If q-p = 24m for any m >= 0 and primes p < q, then p*q-1 is divisible by 24. This follows algebraically from the known "rule" that p^2 - 1 is divisible by 24 for any prime p > 3 as given in A024702.
No result will occur twice, even when including A024702, because the product of any two primes is unique within the set.
Integer results have a density of about 12% to 13% for all possible p,q pairs among the first few hundred primes.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
(5*29-1)/24 = 6, (7*31-1)/24 = 9, (5*53-1)/24 = 11; also note about these three examples, in order, that 29-5 = 24, 31-7 = 24 and 53-5 = 48.
PROG
(PARI) is(n)=my(f=factor(24*n+1)); #f[, 1]==2&&f[1, 2]==1&&f[2, 2]==1 \\ Charles R Greathouse IV, May 30 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Richard R. Forberg, May 27 2013
EXTENSIONS
Missing a(8) from Charles R Greathouse IV, May 31 2013
STATUS
approved