OFFSET
1,1
COMMENTS
Equivalently: k, k+1 and k+2 all have 4 divisors.
There cannot be four consecutive squarefree numbers as one of them is divisible by 2^2 = 4.
These 3 consecutive squarefree numbers of the form p*q have altogether 6 prime factors always including 2 and 3. E.g., if k = 99985, the six prime factors are {2,3,5,19997,33329,49993}. The middle term is even and not divisible by 3.
Nonsquare terms of A056809. First terms of A056809 absent here are A056809(4)=121=11^2, A056809(14)=841=29^2, A056809(55)=6241=79^2.
Cf. A179502 (Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes). - Zak Seidov, Oct 27 2015
The numbers k, k+1, k+2 have the form 2p-1, 2p, 2p+1 where p is an odd prime. A195685 gives the sequence of odd primes that generates these maximal runs of three consecutive integers with four positive divisors. - Timothy L. Tiffin, Jul 05 2016
a(n) is always 1 or 9 mod 12. - Charles R Greathouse IV, Mar 19 2022
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B18.
David Wells, Curious and interesting numbers, Penguin Books, 1986, p. 114.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Roberto Conti, Pierluigi Contucci, and Vitalii Iudelevich, Bounds on tree distribution in number theory, arXiv:2401.03278 [math.NT], 2024. See page 13.
FORMULA
EXAMPLE
33, 34 and 35 all have 4 divisors.
85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.
MATHEMATICA
lst = {}; Do[z = n^3 + 3*n^2 + 2*n; If[PrimeOmega[z/n] == PrimeOmega[z/(n + 2)] == 4 && PrimeNu[z] == 6, AppendTo[lst, n]], {n, 1, 5601, 2}]; lst (* Arkadiusz Wesolowski, Dec 11 2011 *)
okQ[n_]:=Module[{cl={n, n+1, n+2}}, And@@SquareFreeQ/@cl && Union[ DivisorSigma[ 0, cl]]=={4}]; Select[Range[1, 6001, 2], okQ] (* Harvey P. Dale, Dec 17 2011 *)
SequencePosition[DivisorSigma[0, Range[6000]], {4, 4, 4}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2017 *)
PROG
(Haskell)
a039833 n = a039833_list !! (n-1)
a039833_list = f a006881_list where
f (u : vs@(v : w : xs))
| v == u+1 && w == v+1 = u : f vs
| otherwise = f vs
-- Reinhard Zumkeller, Aug 07 2011
(PARI) is(n)=n%4==1 && factor(n)[, 2]==[1, 1]~ && factor(n+1)[, 2]==[1, 1]~ && factor(n+2)[, 2]==[1, 1]~ \\ Charles R Greathouse IV, Aug 29 2016
(PARI) is(n)=my(t=n%12); if(t==1, isprime((n+2)/3) && isprime((n+1)/2) && factor(n)[, 2]==[1, 1]~, t==9 && isprime(n/3) && isprime((n+1)/2) && factor(n+2)[, 2]==[1, 1]~) \\ Charles R Greathouse IV, Mar 19 2022
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Additional comments from Amarnath Murthy, Vladeta Jovovic, Labos Elemer and Benoit Cloitre, May 08 2002
STATUS
approved