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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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37
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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 Elemer, 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.
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LINKS
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Eric Weisstein's World of Mathematics, Squareful
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FORMULA
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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)
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EXAMPLE
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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}.
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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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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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EXTENSIONS
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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.
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STATUS
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approved
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