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

1,1

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

References

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

Links

Table of n, a(n) for n=1..18.

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

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

Example

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}.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A249572 A078236 A054881 * A051681 A267987 A056407

Adjacent sequences: A045879 A045880 A045881 * A045883 A045884 A045885

Keyword

nonn,nice,hard,more

Author

Erich Friedman

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.

Status

approved