A029803 -id:A029803 - 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.

A297265 Numbers whose base-8 digits have equal down-variation and up-variation; see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 65, 73, 81, 89, 97, 105, 113, 121, 130, 138, 146, 154, 162, 170, 178, 186, 195, 203, 211, 219, 227, 235, 243, 251, 260, 268, 276, 284, 292, 300, 308, 316, 325, 333, 341, 349, 357, 365, 373, 381, 390, 398, 406

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.

a(n) = A029803(n+1) for 1 <= n < 72, but a(72) = 521 differs from A029803(73) = 585. - Georg Fischer, Oct 30 2018

			406 in base-8:  6,2,6, having DV = 4, UV = 4, so that 406 is in the sequence.

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

Cf. A297330, A297264, A297266.

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 = 8; 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]   (* A297264 *)
Take[Flatten[Position[w, 0]], 120]    (* A297265 *)
Take[Flatten[Position[w, 1]], 120]    (* A297266 *)

A256088 Non-palindromic balanced numbers in base 8.

Original entry on oeis.org

536, 608, 680, 706, 752, 778, 824, 850, 899, 922, 971, 994, 1049, 1072, 1121, 1144, 1193, 1219, 1265, 1291, 1337, 1363, 1412, 1435, 1484, 1507, 1562, 1585, 1634, 1657, 1706, 1732, 1778, 1804, 1850, 1876, 1925, 1948, 1997

Offset: 1

M. F. Hasler, Mar 14 2015

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero. Since palindromes (A029803) are trivially balanced, they are excluded here.

This is the base-8 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A256082 - A256089, A256075, A256080, A029803.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 8);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=8,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

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]] &]

A319584 Numbers that are palindromic in bases 2, 4, and 8.

Original entry on oeis.org

0, 1, 3, 5, 63, 65, 195, 325, 341, 4095, 4097, 4161, 12291, 12483, 20485, 20805, 21525, 21845, 258111, 262143, 262145, 266305, 786435, 798915, 1310725, 1311749, 1331525, 1332549, 1376277, 1377301, 1397077, 1398101, 16515135, 16777215, 16777217, 16781313

Offset: 1

Jeremias M. Gomes, Sep 23 2018

Intersection of A006995, A014192, and A029803.
From A.H.M. Smeets, Jun 08 2019: (Start)
Intersection of A006995 and A259382.
Intersection of A014192 and A259380.
Intersection of A029803 and A097856.
All repunit numbers in base 2 with 6*k digits are included in this sequence, i.e., all terms A000225(6*k) for k >= 0.
All repunit numbers in base 4 with 2+3*k digits are included in this sequence, i.e., all terms A002450(2+3*k) for k >= 0.
All terms A000051(6*k) for k > 0 are included in this sequence.
All terms A052539(3*k) for k > 0 are included in this sequence.
In general, for sequences with palindromic numbers in the set of bases {b, b^2, ..., b^k}, gaps of size 2 occur at the term pairs (b^(k!) - 1, b^(k!) + 1). See also A319598 for b = 2 and k = 4.
The terms occur in bursts with large gaps in between as shown in the scatterplots of log_b(a(n)-a(n-1)) versus log_b(n) and log_b(1-a(n-1)/a(n)) versus log_b(n). Terms of this sequence are those with b = 2 and k = 3. For comparison, terms with b = 3 and k = 3 are also shown in these plots.
(End)

			89478485 = 101010101010101010101010101_2 = 11111111111111_4 = 525252525_8.

A.H.M. Smeets, Table of n, a(n) for n = 1..2298
A.H.M. Smeets, Scatterplot of log_b(a(n)-a(n-1)) versus log_b(n)
A.H.M. Smeets, Scatterplot of log_b(1-a(n-1)/a(n)) versus log_b(n)

Cf. A006995 (base 2), A014192 (base 4), A029803 (base 8), A097956 (bases 2 and 4), A259380 (bases 2 and 8), A259382 (bases 4 and 8), A319598 (bases 2, 4, 8 and 16).

Cf. A000051, A000225, A002450, A052539.

[n: n in [0..2*10^7] | Intseq(n, 2) eq Reverse(Intseq(n, 2)) and Intseq(n, 4) eq Reverse(Intseq(n, 4)) and Intseq(n, 8) eq Reverse(Intseq(n, 8))]; // Vincenzo Librandi, Sep 24 2018

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 4] && palQ[n, 8], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)
Select[Range[0,168*10^5],AllTrue[Table[IntegerDigits[#,d],{d,{2,4,8}}],PalindromeQ]&] (* Harvey P. Dale, Jan 27 2024 *)

ispal(n, b) = my(d=digits(n, b)); Vecrev(d) == d;
isok(n) = ispal(n, 2) && ispal(n, 4) && ispal(n, 8); \\ Michel Marcus, Jun 11 2019

def nextpal(n, base): # m is the first palindrome successor of n in base base
    m, pl = n+1, 0
    while m > 0:
        m, pl = m//base, pl+1
    if n+1 == base**pl:
        pl = pl+1
    n = n//(base**(pl//2))+1
    m, n = n, n//(base**(pl%2))
    while n > 0:
        m, n = m*base+n%base, n//base
    return m
def rev(n, b):
    m = 0
    while n > 0:
        n, m = n//b, m*b+n%b
    return m
n, a = 1, 0
while n <= 100:
    if a == rev(a, 4) == rev(a, 2):
        print(a)
        n += 1
    a = nextpal(a, 8) # A.H.M. Smeets, Jun 08 2019

[n for n in (0..1000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(4)).is_palindrome() and Word(n.digits(8)).is_palindrome()]

A319598 Numbers in base 10 that are palindromic in bases 2, 4, 8, and 16.

Original entry on oeis.org

0, 1, 3, 5, 4095, 4097, 12291, 20485, 21845, 16777215, 16777217, 16781313, 50331651, 50343939, 83886085, 83906565, 89458005, 89478485, 68702703615, 68719476735, 68719476737, 68736258049, 206158430211, 206208774147, 343597383685, 343602954245, 343681290245

Offset: 1

Jeremias M. Gomes, Sep 24 2018

Intersection of A006995, A014192, A029803, and A029730.

This sequence is infinite because it contains terms of the forms 4096^k-1 (k>=0) and 4096^k+1 (k>0). - Bruno Berselli, Sep 24 2018

			4095 = 111111111111_2 = 333333_4 = 7777_8 = FFF_16. Hence 4095 is in the sequence.

A.H.M. Smeets, Table of n, a(n) for n = 1..114 (first 70 terms from Rémy Sigrist)

Cf. A006995 (base 2), A014192 (base 4), A029803 (base 8), and A029730 (base 16).

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 4] && palQ[n, 8] && palQ[n, 16], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

[n for n in (0..100000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(4)).is_palindrome() and Word(n.digits(8)).is_palindrome() and Word(n.digits(16)).is_palindrome()]

a(19)-a(27) from Rémy Sigrist, Nov 15 2018

A046240 Cubes which are palindromes in base 8.

Original entry on oeis.org

0, 1, 27, 729, 274625, 389017, 135005697, 68769820673, 72043225281, 35187593412609, 18014604668698625, 18120364883707393, 9223385231000600577, 4722367327294625677313, 4725826936714463031297

Offset: 1

Patrick De Geest, May 15 1998

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

Intersection of A029803 and A000578.

Cf. A046239.

a(n) = A046239(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.

A043026 Base-8 palindromes that start with 6.

Original entry on oeis.org

6, 54, 390, 398, 406, 414, 422, 430, 438, 446, 3078, 3150, 3222, 3294, 3366, 3438, 3510, 3582, 24582, 24646, 24710, 24774, 24838, 24902, 24966, 25030, 25102, 25166, 25230, 25294, 25358, 25422, 25486, 25550, 25622, 25686

Offset: 1

Clark Kimberling

Cf. A007094, A029803, A043070.

cp's OEIS Frontend

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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A297265 Numbers whose base-8 digits have equal down-variation and up-variation; see Comments.

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A256088 Non-palindromic balanced numbers in base 8.

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A333423 Numbers that are palindromes in primorial base.

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A319584 Numbers that are palindromic in bases 2, 4, and 8.

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A319598 Numbers in base 10 that are palindromic in bases 2, 4, 8, and 16.

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A046240 Cubes which are palindromes in base 8.

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