

A039833


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


16



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
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OFFSET

1,1


COMMENTS

Equivalently: n, n+1 and n+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 form pq have altogether 6 prime factors always including 2 and 3. E.g., if n=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 n with property that n^2, n^2+1 and n^2+2 are all semiprimes).  Zak Seidov, Oct 27 2015
The numbers n, n+1, n+2 have the form 2p1, 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


REFERENCES

D. Wells, Curious and interesting numbers, Penguin Books.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000


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 !! (n1)
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


CROSSREFS

Subsequence of A006881 and A056809.
Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A195685, A179502.
Sequence in context: A052214 A063838 A075039 * A250732 A080700 A292366
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



