A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Author

Erich Friedman

Keywords

• nonn
• nice
• hard
• more

Comments

Solution for n=10 is same as for n=11.

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

Examples

			a(3) = 48 as 48, 49 and 50 are divisible by squares.
n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

Links

• L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the author page.
• "sikefield3", double-square (2019)
• Eric Weisstein's World of Mathematics, Squarefree numbers
• Eric Weisstein's World of Mathematics, Squareful

Crossrefs

Cf. A013929, A053806, A049535, A077647, A078143. Also A069021 and A051681 are different versions.

Programs

Mathematica

cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]

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

Formula

a(n) = 1 + A020754(n+1). - R. J. Mathar, Jun 25 2010

Correction from Jeppe Stig Nielsen, Mar 05 2022: (Start)

a(n) = 1 + A020754(n+1) for 1 <= n < 11.

a(n) = 1 + A020754(n) for 11 <= n < N where N is unknown.

Possibly a(n) = 1 + A020754(n-d) for some higher n, depending on how many repeated terms the sequence has. (End)

a(n) <= A061742(n) = A002110(n)^2 is the trivial bound obtained from the CRT. - Charles R Greathouse IV, Sep 06 2022

Extensions

a(9)-a(11) from Patrick De Geest, Nov 15 1998, Jan 15 1999

a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999

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]

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]

a(18) was obtained as a result of a team effort by L. Marmet et al.