A279092 -id:A279092 - cp's OEIS frontend

A279093 Numbers that are nontrivially palindromic in three or more consecutive integer bases.

178, 300, 373, 676, 1111, 1702, 2473, 3448, 4651, 6106, 7837, 9868, 12223, 14926, 18001, 21472, 25363, 29698, 34501, 39796, 45607, 51958, 58873, 66376, 74491, 83242, 92653, 102748, 113551, 125086, 137377, 150448, 164323, 179026, 194581, 211012, 228343, 246598

Offset: 1

Jon E. Schoenfield, Jan 31 2017

For any integer b > 1, the base-b expansion of any number k < b will be a one-digit number, and will thus be trivially palindromic.
For each j >= 5 and odd, k = (j^3 + 6*j^2 + 14*j + 11)/2 is a term in the sequence, and represents a 3-digit palindrome in each of three consecutive integer bases:
.
  base  1st digit  2nd digit  3rd digit
  ----  ---------  ---------  ---------
   j+1   (j+3)/2    (j+5)/2    (j+3)/2
   j+2   (j+1)/2    (j+3)/2    (j+1)/2
   j+3   (j-1)/2    (j+7)/2    (j-1)/2
.
(see 178 and 373 in the Example section). Nearly all of the first 95 terms of this sequence are terms of this form.
For each j >= 44 and divisible by 4, k = (3*j^5 + 30*j^4 + 125*j^3 + 270*j^2 + 307*j + 148)/4 is a term in the sequence, and represents a 5-digit palindrome in each of three consecutive integer bases:
.
  base  1st digit  2nd digit  3rd digit  4th digit  5th digit
  ----  ---------  ---------  ---------  ---------  ---------
   j+1  3*j/4 + 4   j/2 +  9   j/4 + 11   j/2 +  9  3*j/4 + 4
   j+2  3*j/4 + 1   j/2 +  2   j/4 +  0   j/2 +  2  3*j/4 + 1
   j+3  3*j/4 - 2   j/2 + 10   j/4 - 11   j/2 + 10  3*j/4 - 2
.
[Reformatted by Jon E. Schoenfield, Apr 01 2018]
From Matej Veselovac, Mar 31 2018: (Start)
Similarly to the one 3-digit and one 5-digit families given above, at least seven more infinite families exist, for 7-digit consecutive palindromes. Given a nonnegative integer n, we have the following representations palindromic in exactly three consecutive integer number bases j+1, j+2, j+3 :
1. For each j = 36+12n, k = (816 + 2474*j + 3114*j^2 + 2117*j^3 + 852*j^4 + 209*j^5 + 30*j^6 + 2*j^7)/12 is a term of the sequence.
2. For each j = 55+6n, k = (245 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/6 is a term of the sequence.
3. For each j = 73+2n, k = (247 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/2 is a term of the sequence.
4. For each j = 116+12n, k = (2440 + 7366 j + 9694 j^2 + 7171 j^3 + 3232 j^4 + 895 j^5 + 142 j^6 + 10 j^7)/12 is a term of the sequence.
5. For each j = 172+6n, k = (812 + 2446 j + 3290 j^2 + 2527 j^3 + 1190 j^4 + 343 j^5 + 56 j^6 + 4 j^7)/6 is a term of the sequence.
6. For each j = 288+12n, k = (1176 + 3566 j + 4374 j^2 + 2807 j^3 + 1032 j^4 + 227 j^5 + 30 j^6 + 2 j^7)/12 is a term of the sequence.
7. For each j = 277+6n, k = (1237 + 3740 j + 4900 j^2 + 3590 j^3 + 1600 j^4 + 440 j^5 + 70 j^6 + 5 j^7)/6 is a term of the sequence.
The smallest terms given by these families are of magnitudes ~ 10^10.3, 10^11.5, 10^12.8, 10^14.4, 10^15.5, 10^16.4 and 10^17. The smallest term of the next family, if it exists, is at least of magnitude ~ 10^18.
Almost all known terms of the sequence so far belong in one of the above defined families, either being 3-, 5-, or 7- digit palindromes in exactly 3 consecutive integer number bases.
There are 13 known terms that do not belong to any families: 300, 3360633, 19987816, 43443858, 532083314, 1778140759, 2721194733, 11325719295, 47622367425, 97638433343, 224678540182, 265282702996, 561091062285 (all but 300 so far are 7-digit cases).
Infinite families for consecutive palindromes longer than 7 digits, as well as any examples for those cases, have not yet been observed.
Smallest example for 9-digit consecutive palindromes does not exist within first 100 integer number bases, thus is at least > 10^16.
Similarly, no terms palindromic in 4 or more consecutive integer number bases have been found, so far.
[Extended by Matej Veselovac, Feb 05 2019] (End)

			178 is in the sequence because the bases in which 178 is nontrivially palindromic include 6, 7, and 8: 178 = 454_6 = 343_7 = 262_8.
373 is in the sequence because the bases in which 373 is nontrivially palindromic include 8, 9, and 10: 373 = 565_8 = 454_9 = 373_10.
265282702996 is in the sequence because the bases in which it is nontrivially palindromic include 43, 44, and 45.
130 is nontrivially palindromic in 7 integer bases (11211_3 = 2002_4 = 202_8 = aa_12 = 55_25 = 22_64 = 11_129), but these bases do not include three consecutive integers, so 130 is not in the sequence.

Matej Veselovac, Table of n, a(n) for n = 1..10000
Matej Veselovac, First seven 7-digit families: digits of palindromes

Cf. A002113 (palindromes in base 10), A048268 (smallest palindrome greater than n in bases n and n+1).
Numbers that are palindromic in bases k and k+1: A060792 (k=2), A097928 (k=3), A097929 (k=4), A097930 (k=5), A097931 (k=6), A099145 (k=7), A099146 (k=8), A029965 (k=9), A029966 (k=11).
Cf. A279092 (numbers that are nontrivially palindromic in two or more consecutive integer bases).

A327810 Numbers that are nontrivially palindromic in two or more consecutive integer bases with non-constant number of digits .

Original entry on oeis.org

10, 130, 651, 2997, 6643, 6886, 9222, 11950, 26691, 27741, 76449, 175850, 193662, 626626, 704396, 723296, 749470, 755846, 883407, 1181729, 1422773, 1798303, 1817179, 2347506, 2593206, 4252232, 5415589, 10453500, 11435450, 17099841, 17402241, 25651017

Offset: 1

Matej Veselovac, Sep 26 2019

This is a subsequence of A279092. That is, each term of A279092 either has an equal number of digits in all the corresponding consecutive bases, or does not, in which case it belongs to this sequence.
Most numbers in A279092 do not belong to this sequence. That is, "unbalanced" consecutive palindromes are much rarer than "balanced" consecutive palindromes.
Specifically, any subsequence of this sequence that is given by fixing the maximum allowed number of digits in the consecutive bases seems to be finite. In contrast, every such subsequence of A279092 is known to be infinite.

			Number 10 can be written as 2*4^1+2*4^0=(2,2)_{4} in base four as a palindrome, and as 1*3^2+0*3^1+1*3^0=(1,0,1)_{3} in base three as a palindrome. The bases 4,3 are consecutive, and have 2,3 digits in their representations respectively. All of this makes the number 10=a(1) a term of the sequence.
a(1)  =10    =(2,2)_{4}               =(1,0,1)_{3}
a(2)  =130   =(2,0,0,2)_{4}           =(1,1,2,1,1)_{3}
a(3)  =651   =(3,0,0,3)_{6}           =(1,0,1,0,1)_{5}
a(4)  =2997  =(5,6,6,5)_{8}           =(1,1,5,1,1)_{7}
a(5)  =6643  =(1,0,0,0,1,0,0,0,1)_{3} =(1,1,0,0,1,1,1,1,1,0,0,1,1)_{2}
a(6)  =6886  =(6,8,8,6)_{10}          =(1,0,4,0,1)_{9}
a(7)  =9222  =(2,4,3,3,4,2)_{5}       =(2,1,0,0,0,1,2)_{4}
a(8)  =11950 =(2,3,2,2,2,3,2)_{4}     =(1,2,1,1,0,1,1,2,1)_{3}
a(9)  =26691 =(3,2,3,3,2,3)_{6}       =(1,3,2,3,2,3,1)_{5}
a(10) =27741 =(3,3,2,2,3,3)_{6}       =(1,3,4,1,4,3,1)_{5}

Max Alekseyev, Table of n, a(n) for n = 1..74
Math StackExchange, Finitely many palindromes in two consecutive number bases, for fixed and distinct numbers of digits

Cf. A279092, A323742.

c[b1_, d_] := Pick[FromDigits[#, b1 + 1] & /@ #, PalindromeQ[#] && Length[#] > 1 && Length[#] != 2 d + 1 & /@ #] &@IntegerDigits[FromDigits[#, b1] & /@ (Flatten[Outer[List, Range[1, b1 - 1], Sequence @@ ConstantArray[Range[0, b1 - 1], d + 0]], d + 0][[All, Join[Range[d + 1], Reverse[Range[1, d + 0]]]]]), b1 + 1]; a[L_] := DeleteDuplicates[Sort[Select[Join[{10}, Flatten[Table[c[b1, d], {d, 2, Ceiling[Log[2, L]/2] + 1}, {b1, 2, Ceiling[L^(1/(2 d))]}]]], # < L &]]]; a[10^6] (* Matej Veselovac, Sep 28 2019 *)

Edited and terms a(11) onward added by Max Alekseyev, Sep 26 2019

Edited by Max Alekseyev, Jun 14 2020

A308657 Smallest number that is nontrivially palindromic in n consecutive number bases.

Original entry on oeis.org

3, 10, 178

Offset: 1

Matej Veselovac, Jun 14 2019

Nontrivially palindromic means having at least 2 digits in the palindromic base representation.
| n | term | consecutive palindromic bases representations |
+---+------+-----------------------------------------------+
| 1 |   3  | 11_2                                          |
| 2 |  10  | 101_3 = 22_4                                  |
| 3 | 178  | 454_6 = 343_7 = 262_8                         |
It is not known if the fourth term exists. The problem can be looked at in context of Diophantine equations, which seem hard.

			a(1) = 3 because it is the smallest nontrivial palindrome in some number base: 11 when written in binary.
a(2) = A279092(1) = 10 because it is the smallest nontrivial palindrome in two consecutive number bases, namely, bases 3 and 4: 101 and 22 when written in those number bases, respectively.
a(3) = A279093(1) = 178 since it can be written as a palindrome, in three consecutive number bases, and it is the smallest such number. Those bases are 6, 7, 8 and those representations are 454, 343, 262.

Mathoverflow, Can a number be palindromic in more than 3 consecutive number bases?

Cf. A002113 (palindromes in base 10).

Cf. A279092, A279093 (numbers that are nontrivially palindromic in k or more consecutive integer bases with k=2,3; for k>=4, no examples are known).

aQ[n_, m_] := SequenceCount[Length[(d = IntegerDigits[n, #])] > 1 && PalindromeQ[d] & /@ Range[2, Ceiling[Sqrt[n]]], Table[True, {m}]] > 0; a[m_] := Module[{n = 2}, While[!aQ[n, m], n++]; n]; Array[a, 3] (* Amiram Eldar, Jul 19 2019 *)

A333512 Numbers nontrivially palindromic in exactly three consecutive number bases.

Original entry on oeis.org

1654123, 4564873, 1687837537501, 195766180573603

Offset: 1

Matej Veselovac, Mar 25 2020

Numbers which are strictly non-palindromic up to a set of k=3 consecutive number bases. It is conjectured that such a sequence for k>=4 is empty. For a special case of k=0, we have the sequence A016038.
A subsequence of A279093. Notice that a(1),a(2),a(3),a(4) are all of the form (b^3 + 3 b^2 + 5 b + 2)/2 where b=2k+6, for k=71,101,7497,36575. Not all terms are necessarily of this form. (See comments in A279093, containing a total of 9 known such forms that generate numbers palindromic in three consecutive number bases.)
For every n, a(n) should be a prime number.

			N = 1654123 is a palindrome when written in three consecutive number bases b = 148,149,150 and is not a palindrome in any other nontrivial number bases 2 <= b <= N-2. The three palindromic representations are: 1654123 = (75,76,75)_148 = (74,75,74)_149 =  (73,77,73)_150. Hence, this number is a term of the sequence.

Cf. A279092, A279093 (consecutive palindromes), A016038 (strictly non-palindromic numbers), A060792 (palindromes in bases 2 and 3).

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A279093 Numbers that are nontrivially palindromic in three or more consecutive integer bases.

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A327810 Numbers that are nontrivially palindromic in two or more consecutive integer bases with non-constant number of digits .

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A308657 Smallest number that is nontrivially palindromic in n consecutive number bases.

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A333512 Numbers nontrivially palindromic in exactly three consecutive number bases.

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