A070284 Smallest of 4 consecutive numbers each divisible by a square.

242, 844, 845, 1680, 1681, 2888, 2889, 3174, 3624, 3625, 3750, 5046, 5047, 8475, 8523, 8954, 10050, 10827, 10924, 10925, 11322, 13374, 14748, 14749, 15775, 15848, 15849, 16575, 17404, 17405, 19647, 19940, 19941, 20574, 21462

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

Keywords

• nonn

Comments

This sequence has positive density in N; the density is around 0.0025.

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of quadruples of terms = {44100k+29349, 44100k+29350, 44100k+29351, 44100k+29352} = {9(4900k+3261), 25(1764k+1174), 49(900k+599), 4(11025k+7338)}; starting terms in this sequence = {29349, 73449, 117549, ...}; difference = A002110(4)^2 = 210^2. - Labos Elemer, Nov 25 2002

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A070258, A068781, A045882, A049535, A077647, A078143.

Programs

Mathematica

f[n_] := Union[Transpose[FactorInteger[n]][[2]]][[ -1]]; a = 0; b = 1; c = 0; Do[d = f[n]; If[a > 1 && b > 1 && c > 1 && d > 1, Print[n - 3]]; a = b; b = c; c = d, {n, 4, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[60000],4,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)

PARI

is(n)=for(i=n,n+3, if(!issquarefree(n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

Formula

A070284 = { A070258[k] | A070258[k+1] = A070258[k]+1 }. - M. F. Hasler, Feb 01 2016

Extensions

More terms from Robert G. Wilson v, May 09 2002

b-file from Charles R Greathouse IV, Jul 23 2010