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}):
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
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Aug 30 2002
EXTENSIONS
More terms from T. D. Noe, Oct 04 2004
STATUS
approved