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A029952 Palindromic in base 5.

%I A029952 #37 Jun 14 2024 01:49:34
%S A029952 0,1,2,3,4,6,12,18,24,26,31,36,41,46,52,57,62,67,72,78,83,88,93,98,
%T A029952 104,109,114,119,124,126,156,186,216,246,252,282,312,342,372,378,408,
%U A029952 438,468,498,504,534,564,594,624,626,651,676,701,726,756,781,806,831
%N A029952 Palindromic in base 5.
%C A029952 Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 03 2020
%H A029952 T. D. Noe, <a href="/A029952/b029952.txt">Table of n, a(n) for n = 1..10000</a>
%H A029952 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 A029952 Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.
%H A029952 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 A029952 <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.
%F A029952 Sum_{n>=2} 1/a(n) = 2.9200482... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020
%p A029952 # test for palindrome in base b, from _N. J. A. Sloane_, Sep 13 2015
%p A029952 b:=5;
%p A029952 ispal := proc(n) global b; local t1,t2,i;
%p A029952 if n <= b-1 then return(1); fi;
%p A029952 t1:=convert(n,base,b); t2:=nops(t1);
%p A029952 for i from 1 to floor(t2/2) do
%p A029952 if t1[i] <> t1[t2+1-1] then return(-1); fi;
%p A029952 od: return(1); end;
%p A029952 lis:=[]; for n from 0 to 100 do if ispal(n) = 1 then lis:=[op(lis),n]; fi; od: lis;
%t A029952 f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,5], AppendTo[lst,n]], {n,1000}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2009 *)
%t A029952 Select[Range[0,1000],IntegerDigits[#,5]==Reverse[IntegerDigits[#,5]]&] (* _Harvey P. Dale_, Oct 24 2020 *)
%o A029952 (Magma) [n: n in [0..900] | Intseq(n, 5) eq Reverse(Intseq(n, 5))]; // _Vincenzo Librandi_, Sep 09 2015
%o A029952 (PARI) ispal(n,b=5)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 03 2020
%o A029952 (Python)
%o A029952 from gmpy2 import digits
%o A029952 def A029952(n):
%o A029952     if n == 1: return 0
%o A029952     y = 5*(x:=5**(len(digits(n>>1,5))-1))
%o A029952     return int((c:=n-x)*x+int(digits(c,5)[-2::-1]or'0',5) if n<x+y else (c:=n-y)*y+int(digits(c,5)[-1::-1]or'0',5)) # _Chai Wah Wu_, Jun 13 2024
%Y A029952 Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
%Y A029952 Cf. A261917, A261918.
%K A029952 nonn,base,easy
%O A029952 1,3
%A A029952 _Patrick De Geest_