A029955 -id:A029955 - cp's OEIS frontend

A262065 Numbers that are palindromes in base-60 representation.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 122, 183, 244, 305, 366

Offset: 1

Reinhard Zumkeller, Sep 10 2015

			.      n | a(n) |  base 60          n |  a(n) |  base 60
.   -----+------+-----------    ------+-------+--------------
.    100 | 2440 | [40, 40]       1000 | 56415 | [15, 40, 15]
.    101 | 2501 | [41, 41]       1001 | 56475 | [15, 41, 15]
.    102 | 2562 | [42, 42]       1002 | 56535 | [15, 42, 15]
.    103 | 2623 | [43, 43]       1003 | 56595 | [15, 43, 15]
.    104 | 2684 | [44, 44]       1004 | 56655 | [15, 44, 15]
.    105 | 2745 | [45, 45]       1005 | 56715 | [15, 45, 15]
.    106 | 2806 | [46, 46]       1006 | 56775 | [15, 46, 15]
.    107 | 2867 | [47, 47]       1007 | 56835 | [15, 47, 15]
.    108 | 2928 | [48, 48]       1008 | 56895 | [15, 48, 15]
.    109 | 2989 | [49, 49]       1009 | 56955 | [15, 49, 15]
.    110 | 3050 | [50, 50]       1010 | 57015 | [15, 50, 15]
.    111 | 3111 | [51, 51]       1011 | 57075 | [15, 51, 15]
.    112 | 3172 | [52, 52]       1012 | 57135 | [15, 52, 15]
.    113 | 3233 | [53, 53]       1013 | 57195 | [15, 53, 15]
.    114 | 3294 | [54, 54]       1014 | 57255 | [15, 54, 15]
.    115 | 3355 | [55, 55]       1015 | 57315 | [15, 55, 15]
.    116 | 3416 | [56, 56]       1016 | 57375 | [15, 56, 15]
.    117 | 3477 | [57, 57]       1017 | 57435 | [15, 57, 15]
.    118 | 3538 | [58, 58]       1018 | 57495 | [15, 58, 15]
.    119 | 3599 | [59, 59]       1019 | 57555 | [15, 59, 15]
.    120 | 3601 | [1, 0, 1]      1020 | 57616 | [16, 0, 16]
.    121 | 3661 | [1, 1, 1]      1021 | 57676 | [16, 1, 16]
.    122 | 3721 | [1, 2, 1]      1022 | 57736 | [16, 2, 16]
.    123 | 3781 | [1, 3, 1]      1023 | 57796 | [16, 3, 16]
.    124 | 3841 | [1, 4, 1]      1024 | 57856 | [16, 4, 16]
.    125 | 3901 | [1, 5, 1]      1025 | 57916 | [16, 5, 16]  .

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (corrected b-file originally from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Palindromic number
Wikipedia, Sexagesimal
Index entries for sequences related to palindromes

Cf. A262079 (first differences).
Intersection with A002113: A262069.
Corresponding sequences for bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.

import Data.List.Ordered (union)
a262065 n = a262065_list !! (n-1)
a262065_list = union us vs where
   us = [val60 $ bs ++ reverse bs | bs <- bss]
   vs = [0..59] ++ [val60 $ bs ++ cs ++ reverse bs |
          bs <- tail bss, cs <- take 60 bss]
   bss = iterate s [0] where
         s [] = [1]; s (59:ds) = 0 : s ds; s (d:ds) = (d + 1) : ds
   val60 = foldr (\b v -> 60 * v + b) 0

[n: n in [0..600] | Intseq(n, 60) eq Reverse(Intseq(n, 60))]; // Vincenzo Librandi, Aug 24 2016

f[n_, b_]:=Module[{i=IntegerDigits[n, b]}, i==Reverse[i]]; lst={}; Do[If[f[n, 60], AppendTo[lst, n]], {n, 400}]; lst (* Vincenzo Librandi, Aug 24 2016 *)
pal60Q[n_]:=Module[{idn60=IntegerDigits[n,60]},idn60==Reverse[idn60]]; Select[Range[0,400],pal60Q] (* Harvey P. Dale, Nov 04 2017 *)

isok(m) = my(d=digits(m, 60)); d == Vecrev(d); \\ Michel Marcus, Jan 22 2022

from sympy import integer_log
from gmpy2 import digits, mpz
def A262065(n):
    if n == 1: return 0
    y = 60*(x:=60**integer_log(n>>1,60)[0])
    return int((c:=n-x)*x+mpz(digits(c,60)[-2::-1]or'0',60) if nChai Wah Wu, Jun 13-14 2024

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A350993 Triangular numbers that are palindromes in base 9.

Original entry on oeis.org

0, 1, 3, 6, 10, 91, 136, 300, 528, 820, 4560, 7381, 11476, 20910, 42486, 66430, 552826, 581581, 597871, 1664400, 2001000, 3420420, 3444000, 5070520, 5380840, 48427561, 75995956, 132494781, 134553810, 137158203, 159213090, 290585778, 434520460, 435848050, 669615310

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((9^k-1)/2) is a term for all k >= 0 (Wishard, 1931).

Also, A000217((3 + 5*9^k)/2) is a term for all k>=0 (Trigg, 1984).

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 9: 10 = 11_9.
91 is a term since 91 = A000217(13) is a triangular number and also a palindromic number in base 9: 91 = 111_9.

Charles W. Trigg, Mathematical Quickies, McGraw Hill Book Co., 1967, Q112, p. 127.

Amiram Eldar, Table of n, a(n) for n = 1..123
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Charles W. Trigg, Problem 281, The College Mathematics Journal, Vol. 15, No. 4 (1984), p. 346; Palindromic Triangular Numbers in Base Nine, Solution to Problem 281, by Michael Vowe, ibid., Vol. 17, No. 2 (1986), pp. 188-189.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.
G. W. Wishard, Problem 3480, The American Mathematical Monthly, Vol. 38, No. 3 (1931), p. 170; Solution to Problem 3480, by Helen A. Merrill, ibid., Vol. 39, No. 3 (1932), p. 179.

Intersection of A000217 and A029955.
The nonary version of A003098.
Cf. A191681, A350987, A350990, A350991, A350992.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 9]] &]

A297268 Numbers whose base-9 digits have equal down-variation and up-variation; see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 82, 91, 100, 109, 118, 127, 136, 145, 154, 164, 173, 182, 191, 200, 209, 218, 227, 236, 246, 255, 264, 273, 282, 291, 300, 309, 318, 328, 337, 346, 355, 364, 373, 382, 391, 400, 410, 419, 428, 437, 446

Offset: 1

Clark Kimberling, Jan 15 2018

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.

Differs from A029955 first at 739=1011_9 which is not a palindrome in base 9 but has DV(739,9)=UV(793,9) =1. - R. J. Mathar, Jan 23 2018

			446 in base-9:  5,4,5, having DV = 1, UV = 1, so that 446 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330, A297267, A297269.

g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 9; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297267 *)
Take[Flatten[Position[w, 0]], 120]    (* A297268 *)
Take[Flatten[Position[w, 1]], 120]    (* A297269 *)

A333423 Numbers that are palindromes in primorial base.

Original entry on oeis.org

0, 1, 3, 7, 9, 11, 31, 39, 47, 211, 217, 223, 229, 235, 243, 249, 255, 261, 267, 275, 281, 287, 293, 299, 2311, 2347, 2383, 2419, 2455, 2523, 2559, 2595, 2631, 2667, 2735, 2771, 2807, 2843, 2879, 30031, 30061, 30091, 30121, 30151, 30181, 30211, 30247, 30277, 30307

Offset: 1

Amiram Eldar, Mar 20 2020

			3 is a term since its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.
7 is a term since its representation in primorial base is 101 (1 * 3# + 0 * 2# + 1 = 6 + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..2500
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A049345, A235168.

Cf. A002113, A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A046807.

max = 6; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[0, nmax], PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

A043035 Base-9 palindromes that start with 8.

Original entry on oeis.org

8, 80, 656, 665, 674, 683, 692, 701, 710, 719, 728, 5840, 5930, 6020, 6110, 6200, 6290, 6380, 6470, 6560, 52496, 52577, 52658, 52739, 52820, 52901, 52982, 53063, 53144, 53234, 53315, 53396, 53477, 53558, 53639, 53720

Offset: 1

Clark Kimberling

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007095, A043079.
Base-9 palindromes: A029955.
Base-9 palindromes that start with d, for d=1..8: A043028..A043035.
Base-b palindromes that start with 8, for b=9,10: this sequence, A043043.

b9dQ[n_]:=Module[{idn=IntegerDigits[n,9]},idn==Reverse[idn]&&idn[[1]] == 8]; Select[Range[60000],b9dQ] (* Harvey P. Dale, Sep 03 2016 *)

A043030 Base-9 palindromes that start with 3.

Original entry on oeis.org

3, 30, 246, 255, 264, 273, 282, 291, 300, 309, 318, 2190, 2280, 2370, 2460, 2550, 2640, 2730, 2820, 2910, 19686, 19767, 19848, 19929, 20010, 20091, 20172, 20253, 20334, 20424, 20505, 20586, 20667, 20748, 20829, 20910

Offset: 1

Clark Kimberling

Robert P. P. McKone, Table of n, a(n) for n = 1..911

Cf. A007095, A029955, A043074.

Cf. A043038.

okQ[n_] := Module[{idn = IntegerDigits[n, 9]}, First[idn] == 3 && FromDigits[IntegerDigits[n, 9]] == FromDigits[Reverse[idn]]]; Select[Range[20910], okQ] (* Robert P. P. McKone, Aug 22 2021, after Harvey P. Dale in A043038 *)

A046242 Cubes which are palindromes in base 9.

Original entry on oeis.org

0, 1, 8, 1000, 54872, 551368, 753571, 389017000, 282558696328, 293151929707, 205901592625000, 150095482587202888, 150714005985636211, 109419057761904073000, 79766448635933205559048, 79802927193348078816187

Offset: 1

Patrick De Geest, May 15 1998

Patrick De Geest, World!Of Numbers, Palindromic cubes in bases 2 to 17.

Intersection of A029955 and A000578.

Cf. A046241.

Select[Range[0,44*10^6]^3,IntegerDigits[#,9]==Reverse[ IntegerDigits[ #,9]]&] (* Harvey P. Dale, Oct 09 2016 *)

a(n) = A046241(n)^3. - Andrew Howroyd, Aug 10 2024

Offset corrected by Andrew Howroyd, Aug 10 2024

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.

A043034 Base-9 palindromes that start with 7.

Original entry on oeis.org

7, 70, 574, 583, 592, 601, 610, 619, 628, 637, 646, 5110, 5200, 5290, 5380, 5470, 5560, 5650, 5740, 5830, 45934, 46015, 46096, 46177, 46258, 46339, 46420, 46501, 46582, 46672, 46753, 46834, 46915, 46996, 47077, 47158

Offset: 1

Clark Kimberling

Cf. A007095, A043078.
Base-9 palindromes: A029955.
Base-9 palindromes that start with d, for d=1..8: A043028..A043035.
Base-b palindromes that start with 7, for b=8..10: A043027, this sequence, A043042.

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A262065 Numbers that are palindromes in base-60 representation.

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A342725 Numbers that are palindromic in base i-1.

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A350993 Triangular numbers that are palindromes in base 9.

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A297268 Numbers whose base-9 digits have equal down-variation and up-variation; see Comments.

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A333423 Numbers that are palindromes in primorial base.

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A043035 Base-9 palindromes that start with 8.

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A043030 Base-9 palindromes that start with 3.

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A046242 Cubes which are palindromes in base 9.

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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.