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A029954 Palindromic in base 7.

%I A029954 #35 Jun 14 2024 11:35:32
%S A029954 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 A029954 128,135,142,150,157,164,171,178,185,192,200,207,214,221,228,235,242,
%U A029954 250,257,264,271,278,285,292,300,307,314,321,328,335,342,344,400,456
%N A029954 Palindromic in base 7.
%C A029954 Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 03 2020
%H A029954 T. D. Noe, <a href="/A029954/b029954.txt">Table of n, a(n) for n = 1..10000</a>
%H A029954 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 A029954 Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.
%H A029954 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 A029954 <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.
%F A029954 Sum_{n>=2} 1/a(n) = 3.1313768... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020
%t A029954 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 A029954 pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[ Range[0,500],pal7Q] (* _Harvey P. Dale_, Jul 30 2015 *)
%o A029954 (Python)
%o A029954 from gmpy2 import digits
%o A029954 def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
%o A029954     if l > 0:
%o A029954         yield 0
%o A029954         for x in range(1,l+1):
%o A029954             for y in range(b**(x-1),b**x):
%o A029954                 s = digits(y,b)
%o A029954                 yield int(s+s[-2::-1],b)
%o A029954             for y in range(b**(x-1),b**x):
%o A029954                 s = digits(y,b)
%o A029954                 yield int(s+s[::-1],b)
%o A029954 A029954_list = list(palQgen(4,7)) # _Chai Wah Wu_, Dec 01 2014
%o A029954 (Python)
%o A029954 from gmpy2 import digits
%o A029954 from sympy import integer_log
%o A029954 def A029954(n):
%o A029954     if n == 1: return 0
%o A029954     y = 7*(x:=7**integer_log(n>>1,7)[0])
%o A029954     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 A029954 (PARI) ispal(n,b=7)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 03 2020
%Y A029954 Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
%K A029954 nonn,base,easy
%O A029954 1,3
%A A029954 _Patrick De Geest_