A029954 - OEIS

%I #35 Jun 14 2024 11:35:32

%S 0,1,2,3,4,5,6,8,16,24,32,40,48,50,57,64,71,78,85,92,100,107,114,121,

%T 128,135,142,150,157,164,171,178,185,192,200,207,214,221,228,235,242,

%U 250,257,264,271,278,285,292,300,307,314,321,328,335,342,344,400,456

%N Palindromic in base 7.

%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="/A029954/b029954.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) = 3.1313768... (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,7], AppendTo[lst,n]], {n,1000}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2009 *)

%t pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[ Range[0,500],pal7Q] (* _Harvey P. Dale_, Jul 30 2015 *)

%o (Python)

%o from gmpy2 import digits

%o def palQgen(l,b): # generator of palindromes in base b of length <= 2*l

%o     if l > 0:

%o         yield 0

%o         for x in range(1,l+1):

%o             for y in range(b**(x-1),b**x):

%o                 s = digits(y,b)

%o                 yield int(s+s[-2::-1],b)

%o             for y in range(b**(x-1),b**x):

%o                 s = digits(y,b)

%o                 yield int(s+s[::-1],b)

%o A029954_list = list(palQgen(4,7)) # _Chai Wah Wu_, Dec 01 2014

%o (Python)

%o from gmpy2 import digits

%o from sympy import integer_log

%o def A029954(n):

%o     if n == 1: return 0

%o     y = 7*(x:=7**integer_log(n>>1,7)[0])

%o     return int((c:=n-x)*x+int(digits(c,7)[-2::-1]or'0',7) if n<x+y else (c:=n-y)*y+int(digits(c,7)[-1::-1]or'0',7)) # _Chai Wah Wu_, Jun 14 2024

%o (PARI) ispal(n,b=7)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 03 2020

%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_