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A262065 Numbers that are palindromes in base-60 representation.

%I A262065 #39 Feb 16 2025 08:33:27
%S A262065 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A262065 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A262065 49,50,51,52,53,54,55,56,57,58,59,61,122,183,244,305,366
%N A262065 Numbers that are palindromes in base-60 representation.
%H A262065 Chai Wah Wu, <a href="/A262065/b262065.txt">Table of n, a(n) for n = 1..10000</a> (corrected b-file originally from Reinhard Zumkeller)
%H A262065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumber.html">Palindromic Number</a>
%H A262065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A262065 Wikipedia, <a href="http://www.wikipedia.org/wiki/Palindromic_number">Palindromic number</a>
%H A262065 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%H A262065 <a href="/index/Pac#palindromes">Index entries for sequences related to palindromes</a>
%e A262065 .      n | a(n) |  base 60          n |  a(n) |  base 60
%e A262065 .   -----+------+-----------    ------+-------+--------------
%e A262065 .    100 | 2440 | [40, 40]       1000 | 56415 | [15, 40, 15]
%e A262065 .    101 | 2501 | [41, 41]       1001 | 56475 | [15, 41, 15]
%e A262065 .    102 | 2562 | [42, 42]       1002 | 56535 | [15, 42, 15]
%e A262065 .    103 | 2623 | [43, 43]       1003 | 56595 | [15, 43, 15]
%e A262065 .    104 | 2684 | [44, 44]       1004 | 56655 | [15, 44, 15]
%e A262065 .    105 | 2745 | [45, 45]       1005 | 56715 | [15, 45, 15]
%e A262065 .    106 | 2806 | [46, 46]       1006 | 56775 | [15, 46, 15]
%e A262065 .    107 | 2867 | [47, 47]       1007 | 56835 | [15, 47, 15]
%e A262065 .    108 | 2928 | [48, 48]       1008 | 56895 | [15, 48, 15]
%e A262065 .    109 | 2989 | [49, 49]       1009 | 56955 | [15, 49, 15]
%e A262065 .    110 | 3050 | [50, 50]       1010 | 57015 | [15, 50, 15]
%e A262065 .    111 | 3111 | [51, 51]       1011 | 57075 | [15, 51, 15]
%e A262065 .    112 | 3172 | [52, 52]       1012 | 57135 | [15, 52, 15]
%e A262065 .    113 | 3233 | [53, 53]       1013 | 57195 | [15, 53, 15]
%e A262065 .    114 | 3294 | [54, 54]       1014 | 57255 | [15, 54, 15]
%e A262065 .    115 | 3355 | [55, 55]       1015 | 57315 | [15, 55, 15]
%e A262065 .    116 | 3416 | [56, 56]       1016 | 57375 | [15, 56, 15]
%e A262065 .    117 | 3477 | [57, 57]       1017 | 57435 | [15, 57, 15]
%e A262065 .    118 | 3538 | [58, 58]       1018 | 57495 | [15, 58, 15]
%e A262065 .    119 | 3599 | [59, 59]       1019 | 57555 | [15, 59, 15]
%e A262065 .    120 | 3601 | [1, 0, 1]      1020 | 57616 | [16, 0, 16]
%e A262065 .    121 | 3661 | [1, 1, 1]      1021 | 57676 | [16, 1, 16]
%e A262065 .    122 | 3721 | [1, 2, 1]      1022 | 57736 | [16, 2, 16]
%e A262065 .    123 | 3781 | [1, 3, 1]      1023 | 57796 | [16, 3, 16]
%e A262065 .    124 | 3841 | [1, 4, 1]      1024 | 57856 | [16, 4, 16]
%e A262065 .    125 | 3901 | [1, 5, 1]      1025 | 57916 | [16, 5, 16]  .
%t A262065 f[n_, b_]:=Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 60], AppendTo[lst, n]], {n, 400}]; lst (* _Vincenzo Librandi_, Aug 24 2016 *)
%t A262065 pal60Q[n_]:=Module[{idn60=IntegerDigits[n,60]},idn60==Reverse[idn60]]; Select[Range[0,400],pal60Q] (* _Harvey P. Dale_, Nov 04 2017 *)
%o A262065 (Haskell)
%o A262065 import Data.List.Ordered (union)
%o A262065 a262065 n = a262065_list !! (n-1)
%o A262065 a262065_list = union us vs where
%o A262065    us = [val60 $ bs ++ reverse bs | bs <- bss]
%o A262065    vs = [0..59] ++ [val60 $ bs ++ cs ++ reverse bs |
%o A262065           bs <- tail bss, cs <- take 60 bss]
%o A262065    bss = iterate s [0] where
%o A262065          s [] = [1]; s (59:ds) = 0 : s ds; s (d:ds) = (d + 1) : ds
%o A262065    val60 = foldr (\b v -> 60 * v + b) 0
%o A262065 (Magma) [n: n in [0..600] | Intseq(n, 60) eq Reverse(Intseq(n, 60))]; // _Vincenzo Librandi_, Aug 24 2016
%o A262065 (PARI) isok(m) = my(d=digits(m, 60)); d == Vecrev(d); \\ _Michel Marcus_, Jan 22 2022
%o A262065 (Python)
%o A262065 from sympy import integer_log
%o A262065 from gmpy2 import digits, mpz
%o A262065 def A262065(n):
%o A262065     if n == 1: return 0
%o A262065     y = 60*(x:=60**integer_log(n>>1,60)[0])
%o A262065     return int((c:=n-x)*x+mpz(digits(c,60)[-2::-1]or'0',60) if n<x+y else (c:=n-y)*y+mpz(digits(c,60)[::-1]or'0',60)) # _Chai Wah Wu_, Jun 13-14 2024
%Y A262065 Cf. A262079 (first differences).
%Y A262065 Intersection with A002113: A262069.
%Y A262065 Corresponding sequences for bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.
%K A262065 nonn,base,look
%O A262065 1,3
%A A262065 _Reinhard Zumkeller_, Sep 10 2015