A051681 Smallest term of first run of exactly n consecutive integers which are not squarefree.

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Author

Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com)

Keywords

• nice
• nonn
• hard
• more

Examples

			a(5) = 844: 844 = 2^2*211, 845 = 5*13^2, 846 = 2*3^2*47, 847 = 7*11^2, 848 = 2^4*53.
		

References

• a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al.

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.

Crossrefs

Cf. A045882 (another version), A013929, A053806.

Programs

Mathematica

Module[{tb=Table[If[SquareFreeQ[n],0,1],{n,11*10^5}]},Table[ SequencePosition[ tb,PadRight[{},k,1],1][[All,1]],{k,8}]]//Flatten (* The program generates the first 8 terms of the sequence. To generate more, increase the constants for n and k but the program may take a long time to run. *) (* Harvey P. Dale, Mar 24 2022 *)

PARI

iscons(x, n)=if (issquarefree(x-1) && issquarefree(x+n), for (k = 0, n-1, if (issquarefree(x+k), return (0));); return (1);); return (0);
a(n) = {my(x = 1); while (! iscons(x, n), x++); x;} \\ Michel Marcus, Jan 13 2014

Extensions

a(16) reported by Louis Marmet (louis(AT)marmet.org), Jul 24 2000

a(17) was obtained as a result of a team effort by E. Wong et al.

a(18) = 125781000834058568 was obtained as a result of a team effort by L. Marmet et al.