A074172 - OEIS

%I #22 Jan 22 2022 23:36:40

%S 44,75,98,116,147,171,244,332,387,507,548,603,604,724,844,908,931,963,

%T 1075,1083,1251,1324,1412,1467,1556,1587,1675,1772,2523,2524,2636,

%U 2644,2763,3283,3356,3411,3508,3788,3987,4075,4203,4204,4418,4491,4804,4868

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

%C From _Robert Israel_, Dec 06 2018: (Start)

%C There are four forms of terms, for odd primes p,q,r:

%C 4*p where 4*p+1 = q^2*r, r == 1 (mod 4)

%C 2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)

%C p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)

%C p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).

%C Are there infinitely many terms of each type?

%C (End)

%H Robert Israel, <a href="/A074172/b074172.txt">Table of n, a(n) for n = 1..10000</a>

%e 44 is a member as 44 = 2^2*11 and 45 = 3^2*5.

%p filter:= proc(n) local F;

%p F:= map(t -> t[2],ifactors(n)[2]);

%p F = [2,1] or F = [1,2]

%p end proc:

%p A054753:= select(filter, {$1..10000}):

%p sort(convert(A054753 intersect map(`-`,A054753,1),list)); # _Robert Israel_, Dec 06 2018

%t 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 (PARI) isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;

%o isok(n) = isok1(n) && isok1(n+1); \\ _Michel Marcus_, Sep 20 2017

%Y Cf. A054753, A074173, A074174, A178032, A308683, A141621.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Aug 30 2002

%E More terms from _T. D. Noe_, Oct 04 2004