A214425 -id:A214425 - cp's OEIS frontend

A238893 Encoded bases for which A214425(n) is palindromic.

179, 238, 135, 268, 359, 137, 137, 258, 136, 268, 237, 578, 268, 567, 589, 137, 257, 367, 269, 138, 136, 138, 489, 679, 678, 137, 268, 137, 268, 178, 179, 289, 135, 258, 147, 137, 137, 137, 128, 268, 137, 137, 268, 137, 137, 137, 137, 248, 139, 259, 137

Offset: 1

T. D. Noe, Mar 07 2014

The three bases b < c < d are encoded as one number (b-1)*100 + (c-1)*10 + (d-1). Similar to A214427 which tabulates the single base for which A214423(n) is palindromic. The vast majority of these palindromes are for the three bases (2,4,8), which encodes as 137 in this sequence.

			A214425(1) = 9. The number 9 is palindromic in 3 bases: 2, 8, and 10. Hence, a(1) = 179.

T. D. Noe, Table of n, a(n) for n = 1..1234

Cf. A214423, A214425, A214427, A238338.

n = -1; t = {}; While[Length[t] < 51, n++; If[Count[c = Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 3, d = Flatten[Position[c, True]]; AppendTo[t, 100*d[[1]] + 10*d[[2]] + d[[3]]]]]; 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.

A214424 Numbers that are palindromic in exactly two bases b, 2 <= b <= 10.

Original entry on oeis.org

15, 16, 17, 18, 20, 24, 26, 27, 28, 31, 33, 36, 45, 46, 50, 51, 52, 57, 67, 73, 78, 82, 88, 91, 92, 93, 98, 99, 104, 105, 107, 109, 111, 114, 119, 127, 129, 135, 141, 142, 150, 151, 160, 170, 171, 173, 182, 185, 186, 200, 209, 212, 215, 219, 227, 246, 252

Offset: 1

T. D. Noe, Jul 18 2012

Every pair of bases occurs. The pair (2,3), for the number a(732) = 1422773, is the last to occur. Note that 1422773 = 101011011010110110101(2) = 2200021200022(3).

See A238338 for the pairs of bases. - T. D. Noe, Mar 07 2014

			15 is palindromic in bases 2 and 4: 15 = 1111_2 = 33_4.

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 [math.NT], 2014.
Edray Herber Goins, Palindromes in different bases: a conjecture of J. Ernest Wilkins, Integers, Vol. 9 (2009), A55.

Cf. A050813, A214423, A214425, A214426 (palindromic in 0-1 and 3-4 bases).

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

pal(v)=v==Vecrev(v)
is(n)=sum(b=2,10,pal(digits(n,b)))==2 \\ Charles R Greathouse IV, Mar 05 2014

A050812(a(n)) = 2.

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.

A260184 Numbers n written in base 10 that are palindromic in exactly three bases b, 2 <= b <= 10 and not simultaneously bases 2, 4 and 8.

Original entry on oeis.org

9, 10, 21, 40, 55, 80, 85, 100, 130, 154, 164, 178, 191, 203, 235, 242, 255, 257, 273, 282, 292, 300, 328, 400, 455, 585, 656, 819, 910, 2709, 6643, 8200, 14762, 32152, 53235, 74647, 428585, 532900, 1181729, 1405397, 4210945, 5259525, 27711772, 719848917, 43253138565

Offset: 1

Giovanni Resta and Robert G. Wilson v, Jul 17 2015

			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.

Cf. A214426, A214425.

(* see A214425 and set all terms as lst, then *)
gQ[n_] := Count[ palQ[n,#] & /@ {2, 4, 8}, True];
Select[ lst, gQ[#] != 3 &]

The intersection of A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955 & A002113 which yields just three members, not simultaneously bases 2, 4 and 8.

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).

cp's OEIS Frontend

A238893 Encoded bases for which A214425(n) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A214424 Numbers that are palindromic in exactly two bases b, 2 <= b <= 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A260184 Numbers n written in base 10 that are palindromic in exactly three bases b, 2 <= b <= 10 and not simultaneously bases 2, 4 and 8.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs