A059809 -id:A059809 - cp's OEIS frontend

A191279 3-digit half-palindromes.

22, 51, 87, 91, 102, 121, 145, 169, 187, 190, 212, 220, 225, 245, 247, 248, 260, 287, 289, 290, 295, 345, 361, 364, 371, 425, 435, 441, 442, 445, 475, 477, 490, 495, 511, 529, 551, 574, 584, 603, 612, 625, 632, 651, 658, 663, 672, 679, 715, 721, 722, 728, 729

Offset: 1

Vladimir Shevelev, May 29 2011

A positive integer m we call k-digit half-palindrome if there exist two bases 1b=[m_k m(k-1)...m_1]_c, where m_i are digits in both of these bases with the condition m_1>0 and m_k>0 (see SeqFan Discussion list from Mar 03 2011, where we introduced "b,c-palindromes").
Robert Israel showed (see SeqFan Discussion list from the same day) that every number of the form [n+1,n,n]_(2*n+1)is 3-digit half-palindrome with b=2*n+1 and c=2*n+2. Thus the sequence is infinite.
On the other hand, every number of the form [k*n+m+1,0,k*n+m]_(4*k*n+4*m+1), where k>=1,m>=0, is 3-digit half-palindrome with b=4*k*n+4*m+1 and c=4*k*n+4*m+3.

			Let m=22. We have 22=[2 1 1]_3 and 22=[1 1 2]_4. Thus 22, by the definition, is a 3-digit half-palindrome.
Let m=91. We have 91=[3 3 1]_5 and 91 =[1 3 3]_8. Thus 91 is a 3-digit half palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..1300

Cf. A002113, A006995, A059809.

q[n_] := Module[{ans = False, db, dc}, Do[db = IntegerDigits[n, b]; If[Length[db] == 3, Do[dc = IntegerDigits[n, c]; If[Length[dc] == 3 && db == Reverse[dc], ans = True; Break[]], {c, b + 1, n - 1}]], {b, 2, n - 1}]; ans]; Select[Range[1000], q] (* Amiram Eldar, Jun 18 2025 *)

Corrected by R. J. Mathar, Jul 02 2012

More terms from Amiram Eldar, Jun 18 2025

A327225 For any n >= 0, let u and v be such that 2 <= u < v and the digits of n in bases u and v are the same up to a permutation and v is minimized; a(n) = u.

Original entry on oeis.org

2, 2, 3, 4, 5, 6, 7, 3, 9, 4, 11, 5, 13, 4, 15, 7, 5, 5, 19, 6, 21, 5, 3, 7, 25, 6, 6, 13, 4, 9, 7, 7, 33, 8, 8, 11, 7, 7, 7, 19, 13, 13, 10, 10, 7, 7, 5, 9, 49, 9, 8, 5, 4, 10, 13, 13, 9, 9, 9, 19, 61, 10, 10, 10, 9, 9, 5, 9, 6, 13, 11, 11, 73, 10, 9, 12, 9

Offset: 0

Rémy Sigrist, Aug 27 2019

For any n >= 0, the sequence is well defined as the representation of n in any base b >= max(2, n+1) corresponds to a single digit n.

(n, u = A327225(n), v = A327226(n)) = (n, n+1, n+2) iff n = 1 or n is in A059809. - Bernard Schott, Aug 31 2019

			For n = 11:
- the representations of 11 in bases b = 2..9 are:
    b  11 in base b
    -  ------------
    2  "1011"
    3  "102"
    4  "23"
    5  "21"
    6  "15"
    7  "14"
    8  "13"
    9  "12"
- the representation in base 9 is the least that shows the same digits, up to order, to some former base, namely the base 5,
- hence a(11) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A327226 for the corresponding v's.

Cf. A004053, A059809.

a(n) = { my (s=[]); for (v=2, oo, my (d=vecsort(digits(n,v))); if (setsearch(s,d), forstep (u=v-1, 2, -1, if (vecsort(digits(n,u))==d, return (u))), s=setunion(s,[d]))) }

a(n) <= max(2, n+1).

A059808 Numbers which contain the same digits in two different bases.

Original entry on oeis.org

7, 9, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Erich Friedman, Feb 24 2001

			13 can be written as 12 (in base 11) or 21 (in base 6)

Cf. A059809.

isok(n) = {for (b=3, n, db = vecsort(digits(n, b)); for (c = 2, b-1, dc = vecsort(digits(n, c)); if (dc == db, return (1));););} \\ Michel Marcus, Aug 03 2015

A059828 Numbers which contain exactly the same digits (with the correct multiplicity) in 3 different smaller bases.

Original entry on oeis.org

495, 912, 913, 1067, 1650, 2576, 3318, 4293, 4305, 4310, 5055, 6636, 6951, 8184, 8586, 8732, 8772, 10110, 11900, 12467, 12877, 12879, 13129, 15158, 15165, 15222, 15322, 16368, 17172, 17177, 19175, 19321, 19467, 20496, 20506, 21022, 21316

Offset: 1

Erich Friedman, Feb 25 2001

			495 is (2)(12)(1) in base 13, (1)(12)(2) in base 17 and (1)(2)(12) in base 21.

Cf. A059809.

More terms from Sascha Kurz, Mar 05 2002

Offset set to 1 by Michel Marcus, Aug 03 2015

A191303 An infinite sequence of 4-digit half-palindromes.

Original entry on oeis.org

52029, 316725, 1093345, 2811129, 6031029, 11445709, 19879545, 32288625, 49760749, 73515429, 104903889, 145409065, 196645605, 260359869, 338429929, 432865569, 545808285, 679531285, 836439489, 1019069529, 1230089749, 1472300205, 1748632665, 2062150609

Offset: 1

Vladimir Shevelev, May 30 2011

For the definition of k-digit half-palindromes, see A191279. Although there exist infinitely many polynomials taking only 3-digit half-palindrome values (see comment to A191279), only two polynomials are known with all values 4-digit half-palindromes. They are the polynomials P(n) which were discovered in SeqFan Discussion list from Mar 14 2011 and its double.
The sequence lists values of P(n). All these are odd. For a given k>=5, up to now it is unknown if there are polynomials taking only k-digit half-palindrome values and it is unknown whether there exist infinitely many such numbers.
Conjecture. For every k>=2, there exists a polynomial of degree k taking only k-digit half-palindrome values.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002113, A006995, A059809, A191279.

LinearRecurrence[{5,-10,10,-5,1},{52029,316725,1093345,2811129,6031029},20] (* Harvey P. Dale, Sep 19 2018 *)

a(n) = (2*n+2)(14*n+9)^3+(3*n+3)*(14*n+9)^2+(5*n+3)*(14*n+9)+(2*n+1) = (2*n+1)*(14*n+11)^3 +(5*n+3)*(14*n+11)^2 +(3*n+3)*(14*n+11) +(2*n+2), such that the bases b < c are b=14*n+9, c=14*n+11.

G.f. -x*(52029+56580*x+30010*x^2-8636*x^3+1729*x^4) / (x-1)^5. - R. J. Mathar, Jul 01 2012

cp's OEIS Frontend

A191279 3-digit half-palindromes.

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A327225 For any n >= 0, let u and v be such that 2 <= u < v and the digits of n in bases u and v are the same up to a permutation and v is minimized; a(n) = u.

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A059808 Numbers which contain the same digits in two different bases.

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A059828 Numbers which contain exactly the same digits (with the correct multiplicity) in 3 different smaller bases.

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A191303 An infinite sequence of 4-digit half-palindromes.

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