|
%I #46 Dec 01 2022 14:51:44
%S 4,8,48,242,844,22020,217070,1092747,8870024,221167422,221167422,
%T 47255689915,82462576220,1043460553364,79180770078548,
%U 3215226335143218,23742453640900972,125781000834058568
%N Smallest term of first run of (at least) n consecutive integers which are not squarefree.
%C Solution for n=10 is same as for n=11.
%C This sequence is infinite and each term initiates a suitable arithmetic progression with large differences like squares of primorials or other suitable products of primes from prime factors being on power 2 in terms and in chains after. Proof includes solution of linear Diophantine equations and math. induction. See also A068781, A070258, A070284, A078144, A049535, A077640, A077647, A078143 of which first terms are recollected here. - _Labos Elemer_, Nov 25 2002
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
%H L. Marmet, <a href="http://arxiv.org/abs/1210.3829">First occurrences of square-free gaps and an algorithm for their computation</a>, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the <a href="http://www.marmet.org/louis/sqfgap/">author page</a>.
%H "sikefield3", <a href="https://github.com/sikefield3/double-square">double-square</a> (2019)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squareful.html">Squareful</a>
%F a(n) = 1 + A020754(n+1). - _R. J. Mathar_, Jun 25 2010
%F Correction from _Jeppe Stig Nielsen_, Mar 05 2022: (Start)
%F a(n) = 1 + A020754(n+1) for 1 <= n < 11.
%F a(n) = 1 + A020754(n) for 11 <= n < N where N is unknown.
%F Possibly a(n) = 1 + A020754(n-d) for some higher n, depending on how many repeated terms the sequence has. (End)
%F a(n) <= A061742(n) = A002110(n)^2 is the trivial bound obtained from the CRT. - _Charles R Greathouse IV_, Sep 06 2022
%e a(3) = 48 as 48, 49 and 50 are divisible by squares.
%e n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.
%t cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]
%o (PARI) a(n)=my(s);for(k=1,9^99,if(issquarefree(k),s=0,if(s++==n,return(k-n+1)))) \\ _Charles R Greathouse IV_, May 29 2013
%Y Cf. A013929, A053806, A049535, A077647, A078143. Also A069021 and A051681 are different versions.
%K nonn,nice,hard,more
%O 1,1
%A _Erich Friedman_
%E a(9)-a(11) from _Patrick De Geest_, Nov 15 1998, Jan 15 1999
%E a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999
%E a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al. [Louis Marmet (louis(AT)marmet.org), Jul 27 2000]
%E a(17) was obtained as a result of a team effort by E. Wong et al. [Louis Marmet (louis(AT)marmet.org), Jul 13 2001]
%E a(18) was obtained as a result of a team effort by L. Marmet et al.
| 