A029957 -id:A029957 - cp's OEIS frontend

A029956 Numbers that are palindromic in base 11.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 366, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 488, 499

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Cf. A002113 (base 10), A029957 (base 12).

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,11],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal11Q[n_]:=Module[{idn11=IntegerDigits[n,11]},idn11==Reverse[idn11]]; Select[Range[0,500],pal11Q] (* Harvey P. Dale, May 11 2015 *)
Select[Range[0, 500], PalindromeQ[IntegerDigits[#, 11]] &] (* Michael De Vlieger, May 12 2017, Version 10.3 *)

ispal(n,b)=my(tmp,d=log(n+.5)\log(b)-1);while(d,tmp=n%b;n\=b;if(n\b^d!=tmp,return(0));n=n%(b^d);d-=2;);d<0||n%(b+1)==0
is(n)=ispal(n,11) \\ Charles R Greathouse IV, Aug 21 2012

ispal(n,b=11)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 04 2020

from gmpy2 import digits
from sympy import integer_log
def A029956(n):
    if n == 1: return 0
    y = 11*(x:=11**integer_log(n>>1,11)[0])
    return int((c:=n-x)*x+int(digits(c,11)[-2::-1]or'0',11) if nChai Wah Wu, Jun 14 2024

[n for n in (0..499) if Word(n.digits(11)).is_palindrome()] # Peter Luschny, Sep 13 2018

Sum_{n>=2} 1/a(n) = 3.4369816... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029958 Numbers that are palindromic in base 13.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 170, 183, 196, 209, 222, 235, 248, 261, 274, 287, 300, 313, 326, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 510, 523, 536, 549, 562

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,13],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range[0,600],IntegerDigits[#,13]==Reverse[IntegerDigits[#,13]]&] (* Harvey P. Dale, Nov 16 2022 *)

isok(n) = my(d=digits(n, 13)); d == Vecrev(d); \\ Michel Marcus, May 13 2017

from sympy import integer_log
from gmpy2 import digits
def A029958(n):
    if n == 1: return 0
    y = 13*(x:=13**integer_log(n>>1,13)[0])
    return int((c:=n-x)*x+int(digits(c,13)[-2::-1]or'0',13) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.55686013... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029959 Numbers that are palindromic in base 14.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 394, 408, 422, 436, 450, 464, 478, 492, 506, 520, 534, 548, 562, 576, 591

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

			195 is DD in base 14.
196 is 100 in base 14, so it's not in the sequence.
197 is 101 in base 14.

John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 13: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958.

palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[ Range[0, 600], palQ[#, 14] &] (* Harvey P. Dale, Aug 03 2014 *)

isok(n) = Pol(d=digits(n, 14)) == Polrev(d); \\ Michel Marcus, Mar 12 2017

from sympy import integer_log
from gmpy2 import digits
def A029959(n):
    if n == 1: return 0
    y = 14*(x:=14**integer_log(n>>1,14)[0])
    return int((c:=n-x)*x+int(digits(c,14)[-2::-1]or'0',14) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.6112482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029960 Numbers that are palindromic in base 15.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 226, 241, 256, 271, 286, 301, 316, 331, 346, 361, 376, 391, 406, 421, 436, 452, 467, 482, 497, 512, 527, 542, 557, 572, 587, 602, 617

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

John Cerkan, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 14: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958, A029959.

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,15],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range@ 620, PalindromeQ@ IntegerDigits[#, 15] &] (* Michael De Vlieger, May 13 2017, Version 10.3 *)

isok(n) = my(d=digits(n, 15)); d == Vecrev(d); \\ Michel Marcus, May 14 2017

from sympy import integer_log
from gmpy2 import digits
def A029960(n):
    if n == 1: return 0
    y = 15*(x:=15**integer_log(n>>1,15)[0])
    return int((c:=n-x)*x+int(digits(c,15)[-2::-1]or'0',15) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.66254285... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A262065 Numbers that are palindromes in base-60 representation.

Original entry on oeis.org

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

A297277 Numbers whose base-12 digits have equal down-variation and up-variation; see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555

Offset: 1

Clark Kimberling, Jan 16 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 first from A029957 after the zero for 1741 = 1011_12, which is not a palindrome in base 12 but has DV(1741,12) = UV(1741,12) =1. - R. J. Mathar, Jan 23 2018

			555 in base-12:  3,10,3, having DV = 7, UV = 7, so that 555 is in the sequence.

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

Cf. A297330, A297276, A297278.

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 = 12; 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]   (* A297276 *)
Take[Flatten[Position[w, 0]], 120]    (* A297277 *)
Take[Flatten[Position[w, 1]], 120]    (* A297278 *)

A046246 Cubes which are palindromes in base 12.

Original entry on oeis.org

0, 1, 8, 2197, 3048625, 3869893, 5168743489, 8917390455553, 9104453457841, 11355729445333, 15407207327425537, 15757560619152625, 19591506830849941, 26623360029195546625, 26669607878348076097, 33849634498353904789

Offset: 1

Patrick De Geest, May 15 1998

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

Intersection of A029957 and A000578.

Cf. A046245.

a(n) = A046245(n)^3. - Andrew Howroyd, Aug 11 2024

Offset corrected by Andrew Howroyd, Aug 11 2024

A253241 The "Reverse and Add!" problem in base 12: sequence lists the final palindrome number for n, or -1 if no palindrome is ever reached. (Written in base 10.)

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 13, 39, 65, 91, 117, 143, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 507, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 507, 676, 65, 78, 91, 104, 117

Offset: 0

Eric Chen, Apr 07 2015

Is a(n) = -1 possible? All numbers below 100 (decimal 144) reach a palindrome.
a(237) is conjectured to be -1.
A060382 lists the smallest possible Lychrel number in base n.

			a(29) = 91 since (in duodecimal) 25 (decimal 29) + 52 = 77 (decimal 91) and 77 is a palindrome.
a(69) = 507 since (in duodecimal) 59 (decimal 69) + 95 = 132, 132 + 231 = 363 (decimal 507) and 363 is a palindrome.
a(105) = 1885 since (in duodecimal) 89 (decimal 105) + 98 = 165, 165 + 561 = 706, 706 + 607 = 1111 (decimal 1885) and 1111 is a palindrome.

Cf. A061563, A062129, A016016, A060382, A029957.

tol = 1728; r[n_] := FromDigits[Reverse[IntegerDigits[n, 12]], 12]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, n = -1]; n, {n, 0, 144}]

A043271 Sum of the digits of the n-th base 12 palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10

Offset: 1

Clark Kimberling

A029957 (base 12 palindromes)

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A029956 Numbers that are palindromic in base 11.

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A029958 Numbers that are palindromic in base 13.

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A029959 Numbers that are palindromic in base 14.

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A029960 Numbers that are palindromic in base 15.

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A262065 Numbers that are palindromes in base-60 representation.

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A297277 Numbers whose base-12 digits have equal down-variation and up-variation; see Comments.

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A046246 Cubes which are palindromes in base 12.

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