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

1,1

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)

Links

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

Example

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

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

Prog

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

Crossrefs

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

Sequence in context: A348076 A348345 A049103 * A039386 A043209 A043989

Adjacent sequences: A074169 A074170 A074171 * A074173 A074174 A074175

Keyword

nonn

Author

Amarnath Murthy, Aug 30 2002

Extensions

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

Name clarified by Sean A. Irvine, Jan 13 2025

Status

approved