A259374 Palindromic numbers in bases 3 and 5 written in base 10.

0, 1, 2, 4, 26, 52, 1066, 1667, 2188, 32152, 67834, 423176, 437576, 14752936, 26513692, 27711772, 33274388, 320785556, 1065805109, 9012701786, 9256436186, 12814126552, 18814619428, 201241053056, 478999841578, 670919564984, 18432110906024, 158312796835916, 278737550525722

Offset

1,3

Comments

0 is only 0 regardless of the base,

1 is only 1 regardless of the base,

2 on the other hand is also 10 in base 2, denoted as 10_2,

3 is 3 in all bases greater than 3, but is 11_2 and 10_3.

Links

Giovanni Resta, Table of n, a(n) for n = 1..39

A.H.M. Smeets, Scatterplot of log_3(number is palindromic in base 3 and base b) versus b, for b in {2,4,5, 6,7,8,10}

Formula

Intersection of A014190 and A029952.

Example

52 is in the sequence because 52_10 = 202_5 = 1221_3.

Mathematica

(* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 5]; If[ palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=5; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10000000}]; lst (* Vincenzo Librandi, Jul 15 2015 *)

Prog

(Python)
def nextpal(n, b): # returns the palindromic successor of n in base b
....m, pl = n+1, 0
....while m > 0:
........m, pl = m//b, pl+1
....if n+1 == b**pl:
........pl = pl+1
....n = (n//(b**(pl//2))+1)//(b**(pl%2))
....m = n
....while n > 0:
........m, n = m*b+n%b, n//b
....return m
n, a3, a5 = 0, 0, 0
while n <= 20000:
....if a3 < a5:
........a3 = nextpal(a3, 3)
....elif a5 < a3:
........a5 = nextpal(a5, 5)
....else: # a3 == a5
........print(n, a3)
........a3, a5, n = nextpal(a3, 3), nextpal(a5, 5), n+1
# A.H.M. Smeets, Jun 03 2019

Crossrefs

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376, A097930, A182234, A259377, A259378, A249156, A097931, A259380-A259384, A099145, A259385-A259390, A099146, A007632, A007633, A029961-A029964, A029804, A029965-A029970, A029731, A097855, A250408-A250411, A099165, A250412.

Sequence in context: A129894 A028386 A362001 * A155120 A356442 A144691

Adjacent sequences: A259371 A259372 A259373 * A259375 A259376 A259377

Keyword

nonn,base

Author

Eric A. Schmidt and Robert G. Wilson v, Jul 14 2015

Status

approved