A262065 Numbers that are palindromes in base-60 representation.

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, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 122, 183, 244, 305, 366

Offset

1,3

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (corrected b-file originally from Reinhard Zumkeller)

Eric Weisstein's World of Mathematics, Palindromic Number

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Palindromic number

Wikipedia, Sexagesimal

Index entries for sequences related to palindromes

Example

.      n | a(n) |  base 60          n |  a(n) |  base 60
.   -----+------+-----------    ------+-------+--------------
.    100 | 2440 | [40, 40]       1000 | 56415 | [15, 40, 15]
.    101 | 2501 | [41, 41]       1001 | 56475 | [15, 41, 15]
.    102 | 2562 | [42, 42]       1002 | 56535 | [15, 42, 15]
.    103 | 2623 | [43, 43]       1003 | 56595 | [15, 43, 15]
.    104 | 2684 | [44, 44]       1004 | 56655 | [15, 44, 15]
.    105 | 2745 | [45, 45]       1005 | 56715 | [15, 45, 15]
.    106 | 2806 | [46, 46]       1006 | 56775 | [15, 46, 15]
.    107 | 2867 | [47, 47]       1007 | 56835 | [15, 47, 15]
.    108 | 2928 | [48, 48]       1008 | 56895 | [15, 48, 15]
.    109 | 2989 | [49, 49]       1009 | 56955 | [15, 49, 15]
.    110 | 3050 | [50, 50]       1010 | 57015 | [15, 50, 15]
.    111 | 3111 | [51, 51]       1011 | 57075 | [15, 51, 15]
.    112 | 3172 | [52, 52]       1012 | 57135 | [15, 52, 15]
.    113 | 3233 | [53, 53]       1013 | 57195 | [15, 53, 15]
.    114 | 3294 | [54, 54]       1014 | 57255 | [15, 54, 15]
.    115 | 3355 | [55, 55]       1015 | 57315 | [15, 55, 15]
.    116 | 3416 | [56, 56]       1016 | 57375 | [15, 56, 15]
.    117 | 3477 | [57, 57]       1017 | 57435 | [15, 57, 15]
.    118 | 3538 | [58, 58]       1018 | 57495 | [15, 58, 15]
.    119 | 3599 | [59, 59]       1019 | 57555 | [15, 59, 15]
.    120 | 3601 | [1, 0, 1]      1020 | 57616 | [16, 0, 16]
.    121 | 3661 | [1, 1, 1]      1021 | 57676 | [16, 1, 16]
.    122 | 3721 | [1, 2, 1]      1022 | 57736 | [16, 2, 16]
.    123 | 3781 | [1, 3, 1]      1023 | 57796 | [16, 3, 16]
.    124 | 3841 | [1, 4, 1]      1024 | 57856 | [16, 4, 16]
.    125 | 3901 | [1, 5, 1]      1025 | 57916 | [16, 5, 16]  .

Mathematica

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 *)
pal60Q[n_]:=Module[{idn60=IntegerDigits[n, 60]}, idn60==Reverse[idn60]]; Select[Range[0, 400], pal60Q] (* Harvey P. Dale, Nov 04 2017 *)

Prog

(Haskell)
import Data.List.Ordered (union)
a262065 n = a262065_list !! (n-1)
a262065_list = union us vs where
   us = [val60 $ bs ++ reverse bs | bs <- bss]
   vs = [0..59] ++ [val60 $ bs ++ cs ++ reverse bs |
          bs <- tail bss, cs <- take 60 bss]
   bss = iterate s [0] where
         s [] = [1]; s (59:ds) = 0 : s ds; s (d:ds) = (d + 1) : ds
   val60 = foldr (\b v -> 60 * v + b) 0
(Magma) [n: n in [0..600] | Intseq(n, 60) eq Reverse(Intseq(n, 60))]; // Vincenzo Librandi, Aug 24 2016
(PARI) isok(m) = my(d=digits(m, 60)); d == Vecrev(d); \\ Michel Marcus, Jan 22 2022
(Python)
from sympy import integer_log
from gmpy2 import digits, mpz
def A262065(n):
    if n == 1: return 0
    y = 60*(x:=60**integer_log(n>>1, 60)[0])
    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

Crossrefs

Cf. A262079 (first differences).

Intersection with A002113: A262069.

Corresponding sequences for bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.

Sequence in context: A227981 A085736 A265711 * A296765 A055643 A122079

Adjacent sequences: A262062 A262063 A262064 * A262066 A262067 A262068

Keyword

nonn,base,look

Author

Reinhard Zumkeller, Sep 10 2015

Status

approved