A214424 -id:A214424 - cp's OEIS frontend

A238338 Encoded bases for which A214424(n) is palindromic.

13, 26, 13, 47, 28, 46, 24, 17, 25, 14, 19, 47, 17, 34, 68, 13, 24, 46, 45, 17, 46, 28, 49, 28, 56, 14, 45, 19, 45, 37, 16, 48, 59, 46, 14, 18, 15, 56, 59, 26, 36, 29, 25, 37, 69, 28, 28, 56, 47, 68, 58, 29, 35, 17, 78, 48, 49, 68, 19, 27, 79, 46, 59, 37, 28, 17

Offset: 1

T. D. Noe, Mar 07 2014

The two bases b < c are encoded as one number (b-1)*10 + (c-1). Similar to A214427 which tabulates the single base for which A214423(n) is palindromic. In the first 1000 terms, the base pairs (2,4), (2,8), (3,9), and (4,8) are most common, occurring 148, 227, 166, and 94 times, respectively. The base pair (2,3) occurs only once.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Attila Bérczes and Volker Ziegler, On simultaneous palindromes, arXiv 1403.0787, Mar 04 2014
Edray Herber Goins, Palindromes in different bases: a conjecture of J. Ernest Wilkins, Integers, Vol. 9 (2009), A55

Cf. A214423, A214424, A214427.

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[c = Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 2, d = Flatten[Position[c, True]]; AppendTo[t, 10*d[[1]] + d[[2]]]]]; t

A050813 Numbers n not palindromic in any base b, 2 <= b <= 10.

Original entry on oeis.org

19, 39, 47, 53, 58, 69, 75, 76, 79, 84, 87, 90, 94, 95, 96, 102, 103, 106, 108, 110, 115, 116, 120, 122, 132, 133, 134, 137, 139, 140, 143, 144, 147, 149, 152, 155, 158, 159, 163, 167, 168, 169, 174, 175, 176, 177, 179, 180, 183, 184, 187, 188, 193, 196, 198

Offset: 1

Patrick De Geest, Oct 15 1999

T. D. Noe, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic numbers beyond base 10

Cf. A050812, A047811, A016038.

Cf. A214423, A214424, A214425, A214426 (palindromic in 1-4 bases).

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 0, AppendTo[t, n]]]; t (* T. D. Noe, Jul 18 2012 *)

A050812(n) = 0.

A214423 Numbers n palindromic in only one base b, 2 <= b <= 10.

Original entry on oeis.org

11, 12, 13, 14, 22, 23, 25, 29, 30, 32, 34, 35, 37, 38, 41, 42, 43, 44, 48, 49, 54, 56, 59, 60, 61, 62, 64, 66, 68, 70, 71, 72, 74, 77, 81, 83, 86, 89, 97, 101, 112, 113, 117, 118, 123, 124, 125, 126, 128, 131, 136, 138, 145, 146, 148, 153, 156, 157, 161

Offset: 1

T. D. Noe, Jul 18 2012

The base for which n is a palindrome is given in A214427.

			11 is palindromic only in base 10.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A050813, A214424, A214425, A214426 (palindromic in 0, 2-4 bases)

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 1, AppendTo[t, n]]]; t

A050812(n) = 1.

A214425 Numbers n palindromic in exactly three bases b, 2 <= b <= 10.

Original entry on oeis.org

9, 10, 21, 40, 55, 63, 65, 80, 85, 100, 130, 154, 164, 178, 191, 195, 203, 235, 242, 255, 257, 273, 282, 292, 300, 325, 328, 341, 400, 455, 585, 656, 819, 910, 2709, 4095, 4097, 4161, 6643, 8200, 12291, 12483, 14762, 20485, 20805, 21525, 21845, 32152, 53235

Offset: 1

T. D. Noe, Jul 18 2012

In the first 1234 terms, only 28 of the possible 84 triples of bases occur. Does every triple occur eventually? - T. D. Noe, Aug 17 2012

See A238893 for the three bases. By far, the most common bases are (2,4,8). - T. D. Noe, Mar 07 2014 (exception are in A260184. - Giovanni Resta and Robert G. Wilson v, Jul 17 2015).

			10 is palindromic in bases 3, 4, and 9.
273 is in the sequence because 100010001_2 = 101010_3 = 10101_4 = 2043_5 = 1133_6 = 540_7 = 421_8 = 333_9 = 273_10 and three of the bases, namely 2, 4 & 9, yield palindromes. - _Giovanni Resta_ and _Robert G. Wilson v_, Jul 17 2015

T. D. Noe, Table of n, a(n) for n = 1..1234
Rick Regan, Finding numbers that are palindromic in multiple bases
Index entries for sequences related to palindromes

Cf. A050813, A214423, A214424, A214426 (palindromic in 0-2 and 4 bases).

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 3, AppendTo[t, n]]]; t

A050812(n) = 3.

The intersection of A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955 & A002113 which yields just three members. - Giovanni Resta and Robert G. Wilson v, Jul 17 2015

A214426 Numbers n palindromic in exactly four bases b, 2 <= b <= 10.

Original entry on oeis.org

8, 121, 373, 786435

Offset: 1

T. D. Noe, Jul 18 2012

Searched up to 10^18. Rick Regan mentions these four numbers, also found by Bill Beckmann, at the end of his webpage. - T. D. Noe, Aug 18 2012

			8 is palindromic in bases 3, 7, 9, and 10.
121 is palindromic in bases 3, 7, 8, and 10.
373 is palindromic in bases 4, 7, 9, and 10.
786435 is palindromic in bases 2, 4, 7, and 8.

Rick Regan, Finding numbers that are palindromic in multiple bases

Cf. A050813, A214423, A214424, A214425 (palindromic in 0-3 bases).

n = -1; t = {}; While[Length[t] < 4, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 4, AppendTo[t, n]]]; t

A050812(n) = 4.

A214422 Least number k > 9 that is palindromic in exactly n bases b, with 2 <= b <= 10.

Original entry on oeis.org

19, 11, 15, 10, 121

Offset: 0

T. D. Noe, Jul 18 2012

There are no other terms < 10^12. The ultimate goal is to find (probably a very large) k that is palindromic in all 9 bases 2 to 10.

			19 is not palindromic in bases 2 to 10.
11 is palindromic in base 10.
15 is palindromic in bases 2 and 4.
10 is palindromic in bases 3, 4, and 9.
121 is palindromic in bases 3, 7, 8, and 10.

Cf. A050813, A214423, A214424, A214425, A214426 (palindromic in 0-4 bases).

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A238338 Encoded bases for which A214424(n) is palindromic.

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A050813 Numbers n not palindromic in any base b, 2 <= b <= 10.

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A214423 Numbers n palindromic in only one base b, 2 <= b <= 10.

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A214425 Numbers n palindromic in exactly three bases b, 2 <= b <= 10.

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A214426 Numbers n palindromic in exactly four bases b, 2 <= b <= 10.

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A214422 Least number k > 9 that is palindromic in exactly n bases b, with 2 <= b <= 10.

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