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

2, 2, 2, 7, 11, 21, 36, 71, 127, 253, 463, 925, 1717, 3433, 6436, 12871, 24311, 48621, 92379, 184757, 352717, 705433, 1352079, 2704157, 5200301, 10400601, 20058301, 40116601, 77558761, 155117521, 300540196, 601080391, 1166803111

Offset

1,1

Comments

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).

Links

Table of n, a(n) for n=1..33.

Formula

Conjecture: a(n) = binomial(n, floor(n/2))+1 for n>3.

Example

For n=5 the minimum base required is 11, giving 11^5=15aa51 (12^5=248832 in base 10).

Mathematica

a[n_] := Module[{b = 2, d},  While[(d = IntegerDigits[(b + 1)^n, b]) != Reverse[d],   b++  ];  b] ;
Table[a[n], {n, 33}] (* James C. McMahon, Dec 13 2023 after PARI *)

Prog

(Python)
from itertools import count
from sympy.ntheory.factor_ import digits
def ispal(d): return d == d[::-1]
def a(n): return next(b for b in count(2) if ispal(digits((b+1)**n, b)[1:]))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 04 2023
(PARI) a(n) = my(b=2, d); while ((d=digits((b+1)^n, b)) != Vecrev(d), b++); b; \\ Michel Marcus, Dec 05 2023

Crossrefs

Cf. A001405.

Sequence in context: A023573 A138757 A158927 * A121258 A087421 A309574

Adjacent sequences: A367854 A367855 A367856 * A367858 A367859 A367860

Keyword

nonn,base

Author

James Carruthers, Dec 03 2023

Extensions

a(8)-a(33) from Michael S. Branicky, Dec 04 2023

Status

approved