A039833 Smallest of three consecutive squarefree numbers k, k+1, k+2 of the form p*q where p and q are distinct primes.

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285, 4413, 4533, 4593, 4881, 5601

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

A008966(a(n)) * A064911(a(n)) * A008966(a(n)+1) * A064911(a(n)+1) * A008966(a(n)+2) * A064911(a(n)+2) = 1. - Reinhard Zumkeller, Feb 26 2011

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

Subsequence of A006881 and A056809 and A263990.

Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A195685, A179502.

Sequence in context: A063838 A075039 A359746 * A355711 A250732 A080700

Adjacent sequences: A039830 A039831 A039832 * A039834 A039835 A039836

Keyword

nonn,nice

Author

Olivier Gérard

Extensions

Additional comments from Amarnath Murthy, Vladeta Jovovic, Labos Elemer and Benoit Cloitre, May 08 2002

Status

approved