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A260824
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Least positive integer b such that b^(2^n)+1 is not squarefree.
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2
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OFFSET
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0,1
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COMMENTS
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The smallest square in the factors of b^(2^n)+1 are 2^2, 5^2, 17^2, 17^2, 769^2. - Robert Price, Mar 07 2017; edited by Jeffrey Shallit, May 10 2017
a(8) <= 50104 (corresponding square 10753^2). - Jeffrey Shallit, May 10 2017
Some better bounds than A261117(n): a(9) <= 65863 (factor 13313^2), a(12) <= 265801 (factor 65537^2), a(16) <= 1493667 (factor 1179649^2), a(18) <= 15834352 (factor 7340033^2), a(19) <= 15786037 (factor 23068673^2), a(21) <= 78597313 (factor 230686721^2), a(22) <= 13753565041 (factor 469762049^2), a(23) <= 6276931961 (factor 469762049^2). - Max Alekseyev, Feb 20 2018
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LINKS
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FORMULA
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EXAMPLE
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For n=4, we consider b^16+1. The first time it is not squarefree is for b=392, where 392^16+1 is divisible by 769^2. So a(4)=392.
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PROG
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(PARI) a(n) = for(b=1, 10^42, !issquarefree(b^(2^n)+1) & return(b) );
(Python)
from sympy.ntheory.factor_ import core
def a(n):
b, pow2, t = 1, 2**n, 2
while core(t, 2) == t:
b += 1
t = b**(pow2) + 1
return b
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CROSSREFS
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KEYWORD
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hard,nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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