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A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

%I A074172 #25 Jan 13 2025 20:31:55
%S A074172 44,75,98,116,147,171,244,332,387,507,548,603,604,724,844,908,931,963,
%T A074172 1075,1083,1251,1324,1412,1467,1556,1587,1675,1772,2523,2524,2636,
%U A074172 2644,2763,3283,3356,3411,3508,3788,3987,4075,4203,4204,4418,4491,4804,4868
%N A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.
%C A074172 From _Robert Israel_, Dec 06 2018: (Start)
%C A074172 There are four forms of terms, for odd primes p,q,r:
%C A074172 4*p where 4*p+1 = q^2*r, r == 1 (mod 4)
%C A074172 2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)
%C A074172 p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)
%C A074172 p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).
%C A074172 Are there infinitely many terms of each type?
%C A074172 (End)
%H A074172 Robert Israel, <a href="/A074172/b074172.txt">Table of n, a(n) for n = 1..10000</a>
%e A074172 44 is a member as 44 = 2^2*11 and 45 = 3^2*5.
%p A074172 filter:= proc(n) local F;
%p A074172 F:= map(t -> t[2],ifactors(n)[2]);
%p A074172 F = [2,1] or F = [1,2]
%p A074172 end proc:
%p A074172 A054753:= select(filter, {$1..10000}):
%p A074172 sort(convert(A054753 intersect map(`-`,A054753,1),list)); # _Robert Israel_, Dec 06 2018
%t A074172 lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst
%o A074172 (PARI) isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;
%o A074172 isok(n) = isok1(n) && isok1(n+1); \\ _Michel Marcus_, Sep 20 2017
%Y A074172 Cf. A054753, A074173, A074174, A178032, A308683, A141621.
%K A074172 nonn
%O A074172 1,1
%A A074172 _Amarnath Murthy_, Aug 30 2002
%E A074172 More terms from _T. D. Noe_, Oct 04 2004
%E A074172 Name clarified by _Sean A. Irvine_, Jan 13 2025