A029953 - OEIS

%I #29 Jun 14 2024 11:35:28

%S 0,1,2,3,4,5,7,14,21,28,35,37,43,49,55,61,67,74,80,86,92,98,104,111,

%T 117,123,129,135,141,148,154,160,166,172,178,185,191,197,203,209,215,

%U 217,259,301,343,385,427,434,476,518,560,602,644,651,693,735,777,819

%N Palindromic in base 6.

%C Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 03 2020

%H T. D. Noe, <a href="/A029953/b029953.txt">Table of n, a(n) for n = 1..10000</a>

%H Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://doi.org/10.1090/mcom/3221">Every positive integer is a sum of three palindromes</a>, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, <a href="http://arxiv.org/abs/1602.06208">arXiv preprint</a>, arXiv:1602.06208 [math.NT], 2017.

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.

%H Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://doi.org/10.13140/RG.2.2.23202.79047">Estimates for the Reciprocal Sum of b-adic Palindromes</a>, 2019.

%H <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.

%F Sum_{n>=2} 1/a(n) = 3.03303318... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020

%t f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,6], AppendTo[lst,n]], {n,1000}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2009 *)

%o (Magma) [n: n in [0..900] | Intseq(n, 6) eq Reverse(Intseq(n, 6))]; // _Vincenzo Librandi_, Sep 09 2015

%o (PARI) ispal(n,b=6)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 03 2020

%o (Python)

%o from gmpy2 import digits

%o from sympy import integer_log

%o def A029953(n):

%o     if n == 1: return 0

%o     y = 6*(x:=6**integer_log(n>>1,6)[0])

%o     return int((c:=n-x)*x+int(digits(c,6)[-2::-1]or'0',6) if n<x+y else (c:=n-y)*y+int(digits(c,6)[-1::-1]or'0',6)) # _Chai Wah Wu_, Jun 14 2024

%Y Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

%K nonn,base,easy

%O 1,3

%A _Patrick De Geest_