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A029959
Numbers that are palindromic in base 14.
6
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 394, 408, 422, 436, 450, 464, 478, 492, 506, 520, 534, 548, 562, 576, 591
OFFSET
1,3
COMMENTS
Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020
LINKS
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
FORMULA
Sum_{n>=2} 1/a(n) = 3.6112482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020
EXAMPLE
195 is DD in base 14.
196 is 100 in base 14, so it's not in the sequence.
197 is 101 in base 14.
MATHEMATICA
palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[ Range[0, 600], palQ[#, 14] &] (* Harvey P. Dale, Aug 03 2014 *)
PROG
(PARI) isok(n) = Pol(d=digits(n, 14)) == Polrev(d); \\ Michel Marcus, Mar 12 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029959(n):
if n == 1: return 0
y = 14*(x:=14**integer_log(n>>1, 14)[0])
return int((c:=n-x)*x+int(digits(c, 14)[-2::-1]or'0', 14) if n<x+y else (c:=n-y)*y+int(digits(c, 14)[-1::-1]or'0', 14)) # Chai Wah Wu, Jun 14 2024
CROSSREFS
Palindromes in bases 2 through 13: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958.
Sequence in context: A048311 A043717 A296753 * A297283 A048325 A048338
KEYWORD
nonn,base,easy
STATUS
approved