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

44, 75, 98, 116, 147, 171, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868

Offset: 1

Author

Amarnath Murthy, Aug 30 2002

Keywords

• nonn

Comments

From Robert Israel, Dec 06 2018: (Start)

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

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

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

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

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

Are there infinitely many terms of each type?

(End)

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

filter:= proc(n) local F;
F:= map(t -> t[2],ifactors(n)[2]);
F = [2,1] or F = [1,2]
end proc:
A054753:= select(filter, {$1..10000}):
sort(convert(A054753 intersect map(`-`,A054753,1),list)); # Robert Israel, Dec 06 2018

Mathematica

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

PARI

isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;
isok(n) = isok1(n) && isok1(n+1); \\ Michel Marcus, Sep 20 2017

Extensions

More terms from T. D. Noe, Oct 04 2004

Name clarified by Sean A. Irvine, Jan 13 2025