A014190 -id:A014190 - cp's OEIS frontend

A029960 Numbers that are palindromic in base 15.

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

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.

A350990 Triangular numbers that are palindromes in base 3.

Original entry on oeis.org

0, 1, 10, 28, 91, 820, 7381, 65341, 66430, 597871, 1633528, 5380840, 48339028, 48427561, 139386556, 435848050, 1178284240, 3529890253, 3922632451, 32614707700, 35296517971, 35303692060, 101891588176, 292358957446, 295883935480, 317733228541, 859413596320, 2649105942220

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((3^k-1)/2) is a term for all k >= 0 (Trigg, 1971).

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 3: 10 = 101_3.
28 is a term since 28 = A000217(7) is a triangular number and also a palindromic number in base 3: 36 = 1001_3.

Amiram Eldar, Table of n, a(n) for n = 1..50
Charles W. Trigg, Problem 3413, Problem Department, School Science and Mathematics, Vol. 71, No. 9 (1971), p. 843; Solution to Problem 3413, by Bob Prielipp, Vol. 72, No. 4 (1972), p. 358.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

Intersection of A000217 and A014190.
The ternary version of A003098.
Cf. A003462, A350987, A350991, A350992, A350993.

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

A173019 a(n) is the value of row n in triangle A083093 seen as ternary number.

Original entry on oeis.org

1, 4, 16, 28, 112, 448, 784, 3136, 12301, 19684, 78736, 314944, 551152, 2204608, 8818432, 15432256, 61729024, 242132884, 387459856, 1549839424, 6199180549, 10848875968, 43395503872, 173577055372, 303766932781, 1215067731124

Offset: 0

Michael Thaler (michael_thaler(AT)brown.edu), Nov 07 2010

Previous name was "Pascal's Triangle mod 3 converted to decimal."
If 2|a(n), then 4|a(n).
If 8|a(n), then 16|a(n).
If a(n)=4*a(n-1), then 3 does not divide n.
The first few odd values for a(n) are a(0)=1, a(8)=12301, a(20)=6199180549, a(24)=303766932781.
It appears that, as the terms of A001317 (analogous to this sequence, using binary instead of ternary) can be uniquely represented as products of Fermat numbers, the terms of this sequence can be represented as products from a nontrivial set of numbers. - Thomas Anton, Oct 27 2018
Subsequence of A014190. - Chai Wah Wu, Jul 30 2025

			a(9) = 3^(3^2) + 1 = 19684;
a(8) = (5*19684 - 12)/8 = 12301;
a(10) = 4*19684 = 78736.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.

Cf. A006940 (takes these values and converts them to decimal notation).

Cf. A001317, A007089, A006943, A083093, A014190.

a173019 = foldr (\t v -> 3 * v + t) 0 . map toInteger . a083093_row
-- Reinhard Zumkeller, Jul 11 2013

FromDigits[#, 3] & /@ Table[Mod[Binomial[n, k], 3], {n, 0, 25}, {k, 0, n}] (* Michael De Vlieger, Oct 31 2018 *)

a(n) = my(v = vector(n+1, k, binomial(n, k-1))); fromdigits(apply(x->x % 3, v), 3); \\ Michel Marcus, Nov 21 2018

from math import prod, comb
from gmpy2 import digits
def A173019(n):
    if n==0: return 1
    c, l = '', len(s:=digits(n,3))
    for k in range(m:=n+2>>1):
        t = digits(k,3).zfill(l)
        c += str(prod(comb(int(s[i]),int(t[i]))%3 for i in range(l))%3)
    return int(c+c[m-2+(n&1)::-1],3) # Chai Wah Wu, Jul 30 2025

a(3^n) = 3^(3^n) + 1.
a(3^n) = (8*a((3^n)-1) + 12)/5. [5*a(3^n) = 1200...0012 (base 3), 8*a((3^n)-1) = (22)(1212...2121) = 11222...2202 (base 3).]
For n > 0, a((3^n)+1) = 4*a(3^n) and a((3^n)+2) = 4*a((3^n)+1).
a(n) = Sum_{k=0..n} A083093(n,k) * 3^k. - Reinhard Zumkeller, Jul 11 2013

a(13) and a(19) corrected and name clarified by Tom Edgar, Oct 11 2015

A330313 Add the odd terms and subtract the even ones, the result must always be a palindrome in base 3. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 3, 2, 11, 7, 4, 6, 8, 21, 5, 12, 14, 89, 9, 18, 26, 16, 20, 10, 13, 17, 24, 75, 39, 30, 32, 23, 51, 31, 22, 43, 34, 48, 44, 28, 36, 19, 29, 42, 81, 68, 33, 35, 69, 60, 72, 63, 73, 52, 56, 40, 105, 61, 70, 84, 93, 91, 82, 50, 41, 98, 45, 53, 103, 64, 78, 54, 123, 128, 57, 71, 129

Offset: 1

N. J. A. Sloane, Dec 11 2019

A base 3 analog of A329544. The latter has an exceptionally irregular graph, so it is natural to ask if the graph is more understandable in a smaller base (and base 2 does not work).

Rémy Sigrist, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
Rémy Sigrist, Density plot of the first 5000000 terms
Rémy Sigrist, C++ program for A330313
N. J. A. Sloane, Table of n, a(n), A330314(n), A330314(n)/a(n) for n = 1..1000

Cf. A014190, A329544, A330312, A330314 (running totals).

A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

Original entry on oeis.org

1, 5, 7, 17, 21, 31, 35, 57, 65, 85, 93, 119, 127, 147, 155, 201, 217, 257, 273, 325, 341, 381, 397, 455, 471, 511, 527, 579, 595, 635, 651, 745, 777, 857, 889, 993, 1025, 1105, 1137, 1253, 1285, 1365, 1397, 1501, 1533, 1613, 1645, 1767, 1799, 1879, 1911, 2015

Offset: 1

Amiram Eldar, Jan 30 2020

Numbers of the form 2^(2*m-1) - 1 (A083420) and 2^(2*m) + 1 (A052539) are terms.

			5 is a term since the negabinary representation of -5 is 1111 which is palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002113, A005352, A006995, A014190, A039724, A094202, A212529, A331191, A331891.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[2000], PalindromeQ @ negabin[-#] &]

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]] &]

A077402 Reverse and Add! carried out in base 3; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 0, 1, 0, 2, 1, 0, 2, 3, 0, 4, 1, 2, 0, 1, 2, 0, 3, 4, 0, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 3, 0, 18, 1, 2, 0, 1, 2, 4, 1, 2, 2, 1, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 2, 3, 2, 4, 17, 18, 0, 1, 0, 2, 1, 1, 2, 1, 1, 3, 1, 0, 2, 1, 1, 16, 1, 1, 2, 2, 0, 2, 4, -1, 16, 3, 15, 2, 1, 1, 2, 1, 0, 3, 3, 3, 2, 1, 1, 16, 1

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base-3 analog of A066057 (base 2), A075685 (base 4) and A033665 (base 10). a(103) = -1 is a conjecture (cf. A066450, A077408). For values of n such that presumably a(n) = -1 see A077404.

			17 (decimal) = 122 -> 122 + 221 = 1120 -> 1120 + 211 = 2101 -> 2101 + 1012 = 10120 -> 10120 + 2101 = 12221 (palindrome) = 160 (decimal) requires 4 steps, so a(17) = 4.

Index entries for sequences related to Reverse and Add!

Cf. A014190, A066450, A033665, A066057, A075685, A077404, A077405.

m := 120; stop := 1000; for n := 0 to m do v := -1; c := 0; k := n; while c < stop do d := k; rev := 0; while d > 0 do rev := 3*rev + (d mod 3); d := d div 3; end; if k = rev then v := c; c := stop; else inc(c); k := k + rev; end; end; write(v,","); end;

A321473 Nonnegative numbers whose nonzero digits in ternary expansion are palindromic.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 16, 18, 20, 23, 24, 26, 27, 28, 30, 31, 34, 36, 37, 39, 40, 46, 48, 52, 54, 56, 59, 60, 62, 65, 68, 69, 72, 74, 78, 80, 81, 82, 84, 85, 88, 90, 91, 93, 94, 100, 102, 106, 108, 109, 111, 112, 117, 118, 120, 121, 130, 136, 138

Offset: 1

Rémy Sigrist, Nov 11 2018

This sequence corresponds to the fixed points of A321464, and contains A014190.

			For n = 1594426:
- the ternary expansion of 1594426 is "10000000010211",
- the corresponding nonzero digits are "11211", which are palindromic,
- hence 1594426 belongs to the sequence.

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

Cf. A014190, A321464.

Select[Range[0,200],PalindromeQ[FromDigits[IntegerDigits[#,3]/.(0-> Nothing)]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2020 *)

is(n, base=3) = my (t=select(sign, digits(n, base))); t==Vecrev(t)

cp's OEIS Frontend

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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A342725 Numbers that are palindromic in base i-1.

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A350990 Triangular numbers that are palindromes in base 3.

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A173019 a(n) is the value of row n in triangle A083093 seen as ternary number.

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A330313 Add the odd terms and subtract the even ones, the result must always be a palindrome in base 3. This is the lexicographically earliest sequence of distinct positive integers with this property.

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A331892 Positive numbers k such that the negabinary expansion (A039724) of -k is palindromic.

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A333423 Numbers that are palindromes in primorial base.

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