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A367857 a(n) is the smallest base b such that (b+1)^n in base b is a palindrome.

%I A367857 #22 Jan 19 2024 14:54:51
%S A367857 2,2,2,7,11,21,36,71,127,253,463,925,1717,3433,6436,12871,24311,48621,
%T A367857 92379,184757,352717,705433,1352079,2704157,5200301,10400601,20058301,
%U A367857 40116601,77558761,155117521,300540196,601080391,1166803111
%N A367857 a(n) is the smallest base b such that (b+1)^n in base b is a palindrome.
%C A367857 Empirically the same as A001405(n)+1 apart from a(2) where 11^10=1001 in base 2 (3^2=9 in base 10) and a(3) where 11^11=11011 in base 2 (3^3=27 in base 10).
%F A367857 Conjecture: a(n) = binomial(n, floor(n/2))+1 for n>3.
%e A367857 For n=5 the minimum base required is 11, giving 11^5=15aa51 (12^5=248832 in base 10).
%t A367857 a[n_] := Module[{b = 2, d},  While[(d = IntegerDigits[(b + 1)^n, b]) != Reverse[d],   b++  ];  b] ;
%t A367857 Table[a[n],{n,33}] (* _James C. McMahon_, Dec 13 2023 after PARI *)
%o A367857 (Python)
%o A367857 from itertools import count
%o A367857 from sympy.ntheory.factor_ import digits
%o A367857 def ispal(d): return d == d[::-1]
%o A367857 def a(n): return next(b for b in count(2) if ispal(digits((b+1)**n, b)[1:]))
%o A367857 print([a(n) for n in range(1, 21)]) # _Michael S. Branicky_, Dec 04 2023
%o A367857 (PARI) a(n) = my(b=2,d); while ((d=digits((b+1)^n, b)) != Vecrev(d), b++); b; \\ _Michel Marcus_, Dec 05 2023
%Y A367857 Cf. A001405.
%K A367857 nonn,base
%O A367857 1,1
%A A367857 _James Carruthers_, Dec 03 2023
%E A367857 a(8)-a(33) from _Michael S. Branicky_, Dec 04 2023