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A260824 Least positive integer b such that b^(2^n)+1 is not squarefree.

%I A260824 #51 Mar 07 2021 09:24:22
%S A260824 3,7,110,40,392,894,315,48
%N A260824 Least positive integer b such that b^(2^n)+1 is not squarefree.
%C A260824 For any n, a(n) <= A261117(n).
%C A260824 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
%C A260824 a(8) <= 50104 (corresponding square 10753^2). - _Jeffrey Shallit_, May 10 2017
%C A260824 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
%F A260824 a(n) = A248214(2^n).
%e A260824 a(1) = A049532(1) = 7.
%e A260824 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.
%o A260824 (PARI) a(n) = for(b=1,10^42, !issquarefree(b^(2^n)+1) & return(b) );
%o A260824 (Python)
%o A260824 from sympy.ntheory.factor_ import core
%o A260824 def a(n):
%o A260824   b, pow2, t = 1, 2**n, 2
%o A260824   while core(t, 2) == t:
%o A260824     b += 1
%o A260824     t = b**(pow2) + 1
%o A260824   return b
%o A260824 print([a(n) for n in range(4)]) # _Michael S. Branicky_, Mar 07 2021
%Y A260824 Subsequence of A248214.
%Y A260824 Cf. A261117, A049532.
%K A260824 hard,nonn,more
%O A260824 0,1
%A A260824 _Jeppe Stig Nielsen_, Aug 04 2015
%E A260824 Edited and a(5)-a(7) added by _Max Alekseyev_, Feb 20 2018