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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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A348076 A348345 A049103 * A039386 A043209 A043989
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Aug 30 2002
EXTENSIONS
More terms from T. D. Noe, Oct 04 2004
STATUS
approved

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Last modified April 14 13:30 EDT 2024. Contains 371661 sequences. (Running on oeis4.)