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A045882
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Smallest term of first run of (at least) n consecutive integers which are not squarefree.
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19
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4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568
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OFFSET
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1,1
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COMMENTS
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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 E. (labos(AT)ana.sote.hu), Nov 25 2002
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829, 2012. - From N. J. A. Sloane, Jan 01 2013
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LINKS
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Table of n, a(n) for n=1..18.
L. Marmet, First occurrences of squarefree gaps...
Eric Weisstein's World of Mathematics, Squarefree numbers
Eric Weisstein's World of Mathematics, Squareful
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FORMULA
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a(n)=1+A020754(n+1). [From R. J. Mathar, Jun 25 2010]
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EXAMPLE
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a(3) = 48 as 48, 49 and 50 are divisible by squares.
E.g. n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.
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MATHEMATICA
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cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]
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CROSSREFS
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Cf. A013929, A053806. Also A069021 and A051681 are different versions.
Cf. A049535, A077647, A078143.
Sequence in context: A078236 A054881 A076687 * A051681 A056407 A056397
Adjacent sequences: A045879 A045880 A045881 * A045883 A045884 A045885
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KEYWORD
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nonn,nice,hard
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AUTHOR
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Erich Friedman
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EXTENSIONS
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a(9) to a(11) from Patrick De Geest, 11/98, 01/99. a(12) to 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) = 125781000834058568 was obtained as a result of a team effort by L. Marmet et al.
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STATUS
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approved
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