cp's OEIS Frontend

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

%I A226133 #14 Jun 16 2023 18:45:11
%S A226133 6,9,11,20,21,23,27,29,30,31,33,34,36,37,38,41,44,45,49,53,56,58,59,
%T A226133 60,61,63,64,65,66,68,79,80,81,82,85,94,96,97,98,102,104,106,107,110,
%U A226133 115,116,120,122,124,128,129
%N A226133 Integers of the form (p*q-1)/24 where p < q are primes.
%C A226133 Results for p = q are given in A024702, which is complementary.
%C A226133 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.
%C A226133 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.
%C A226133 No result will occur twice, even when including A024702, because the product of any two primes is unique within the set.
%C A226133 Integer results have a density of about 12% to 13% for all possible p,q pairs among the first few hundred primes.
%H A226133 Charles R Greathouse IV, <a href="/A226133/b226133.txt">Table of n, a(n) for n = 1..10000</a>
%e A226133 (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.
%o A226133 (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
%Y A226133 Complementary to A024702.
%K A226133 nonn
%O A226133 1,1
%A A226133 _Richard R. Forberg_, May 27 2013
%E A226133 Missing a(8) from _Charles R Greathouse IV_, May 31 2013