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A074172
Smaller of two consecutive numbers of the form p^2*q where p and q are primes.
11
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
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
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Aug 30 2002
EXTENSIONS
More terms from T. D. Noe, Oct 04 2004
STATUS
approved