A226133 Integers of the form (p*q-1)/24 where p < q are primes.

6, 9, 11, 20, 21, 23, 27, 29, 30, 31, 33, 34, 36, 37, 38, 41, 44, 45, 49, 53, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 79, 80, 81, 82, 85, 94, 96, 97, 98, 102, 104, 106, 107, 110, 115, 116, 120, 122, 124, 128, 129

Offset: 1

Author

Richard R. Forberg, May 27 2013

Keywords

• nonn

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.

Examples

			(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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Complementary to A024702.

Programs

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

Extensions

Missing a(8) from Charles R Greathouse IV, May 31 2013