A014190 -id:A014190 - cp's OEIS frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

0, 2, 3, 5, 6, 10, 12, 14, 17, 20, 28, 30, 34, 36, 42, 46, 56, 61, 75, 77, 85, 92, 94, 101, 107, 115, 122, 128, 150, 166, 176, 198, 200, 211, 219, 233, 244, 246, 260, 271, 277, 288, 296, 310, 321, 345, 360, 396, 405, 441, 469, 484, 520, 522, 544, 562, 570, 588

Offset: 1

Amiram Eldar, Feb 17 2022

A001610(n) = Lucas(n+1) - 1 is a term for all n, since A001610(0) = 0 has the representation 0 and the representation of Lucas(n+1) - 1 is n 1's for n > 0.

			The first 10 terms are:
   n  a(n)  A130311(a(n))
   ----------------------
   1   0               0
   2   2               1
   3   3              11
   4   5             101
   5   6             111
   6  10            1111
   7  12           10101
   8  14           11011
   9  17           11111
  10  20          101101

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes.

Cf. A000032, A001610, A130311.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712.

lazy = Select[IntegerDigits[Range[6000], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[# * Reverse @ LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; Join[{0}, Position[s, _?PalindromeQ] // Flatten]

A029971 Palindromic primes in base 3.

Original entry on oeis.org

2, 13, 23, 151, 173, 233, 757, 937, 1093, 1249, 1429, 1487, 1667, 1733, 1823, 1913, 1979, 2069, 8389, 9103, 10111, 12301, 14951, 16673, 16871, 18593, 60103, 60913, 61507, 63127, 69697, 73243, 78979, 80599, 82003, 82813, 83407, 85027

Offset: 1

Patrick De Geest

Intersection of A000040 and A014190. - Michel Marcus, Aug 19 2015

Chai Wah Wu, Table of n, a(n) for n = 1..3004
P. De Geest, World!Of Palindromic Primes

Cf. A117698 (in base 3), A014190.

N:= 14: # to get all terms < 3^N
Res:= 2:
digrev:=proc(n) local L;
  L:= convert(n,base,3);
  add(L[-i]*3^(i-1),i=1..nops(L))
end proc;
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, op(select(isprime,[seq](n*3^m + digrev(n), n=3^(m-1)..3^m-1)));
  else
    m:= (d-1)/2;
    Res:= Res, op(select(isprime,[seq](seq(n*3^(m+1)+y*3^m+digrev(n),
      y=0..2), n=3^(m-1)..3^m-1)));
  fi
od:
Res; # Robert Israel, Aug 19 2015

Do[s = RealDigits[n, 3][[1]]; If[PrimeQ[n], If[FromDigits[s] == FromDigits[Reverse[s]], Print[n]]], {n, 1, 8500}]
Select[Prime[Range[8300]], Reverse[x = IntegerDigits[#, 3]] == x &] (* Jayanta Basu, Jun 23 2013 *)

lista(nn) = forprime(p=2, nn, if ((d=digits(p,3)) && (Vecrev(d)==d), print1(p, ", "))); \\ Michel Marcus, Aug 19 2015

A352087 Numbers whose minimal (or greedy) tribonacci representation (A278038) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 8, 14, 18, 23, 25, 36, 40, 45, 52, 62, 71, 78, 82, 102, 110, 128, 148, 150, 163, 181, 198, 211, 229, 233, 246, 264, 275, 312, 326, 360, 397, 411, 426, 463, 477, 505, 529, 562, 593, 617, 650, 658, 682, 715, 746, 770, 781, 805, 838, 869, 893, 926, 928

Offset: 1

Amiram Eldar, Mar 04 2022

A000073(n) + 1 is a term for n>=4, since its minimal tribonacci representation is 10...01 with n-4 0's between the two 1's.

			The first 10 terms are:
   n  a(n)  A278038(a(n))
  -----------------------
   1   0                0
   2   1                1
   3   3               11
   4   5              101
   5   8             1001
   6  14            10001
   7  18            10101
   8  23            11011
   9  25           100001
  10  36           101101

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

Cf. A000073, A278038.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; PalindromeQ[FromDigits @ IntegerDigits[Total[2^(s - 1)], 2]]]; Select[Range[0, 1000], q]

A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 14, 18, 23, 27, 36, 40, 51, 52, 62, 69, 78, 88, 95, 102, 110, 130, 148, 156, 176, 181, 194, 211, 229, 242, 246, 264, 277, 294, 312, 325, 326, 363, 397, 411, 448, 463, 477, 514, 548, 562, 599, 617, 650, 674, 682, 715, 739, 770, 803, 827, 838, 862

Offset: 1

Amiram Eldar, Mar 05 2022

A027084(n) is a term since its maximal tribonacci representation is n-1 1's and no 0's.

The pairs {A008937(3*k+1)-1, A008937(3*k+1)} = {0, 1}, {7, 8}, {51, 52}, ... are consecutive terms in this sequence: the maximal tribonacci representation of A008937(3*k+1)-1 is 3*k 1's and no 0's (except for k=0 where the representation is 0), and the maximal tribonacci representation of A008937(3*k+1) is of the form 100100...1001 with k blocks of 100 followed by a 1 at the end.

			The first 10 terms are:
   n  a(n)  A352103(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    5             101
   5    7             111
   6    8            1001
   7   14            1111
   8   18           10101
   9   23           11011
  10   27           11111

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

Cf. A000073, A008937, A352103.
A027084 is a subsequence.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; q[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, True, PalindromeQ[FromDigits[v[[i[[1, 1]] ;; -1]]]]]]; Select[Range[0, 1000], q]

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

A214425 Numbers n palindromic in exactly three bases b, 2 <= b <= 10.

Original entry on oeis.org

9, 10, 21, 40, 55, 63, 65, 80, 85, 100, 130, 154, 164, 178, 191, 195, 203, 235, 242, 255, 257, 273, 282, 292, 300, 325, 328, 341, 400, 455, 585, 656, 819, 910, 2709, 4095, 4097, 4161, 6643, 8200, 12291, 12483, 14762, 20485, 20805, 21525, 21845, 32152, 53235

Offset: 1

T. D. Noe, Jul 18 2012

In the first 1234 terms, only 28 of the possible 84 triples of bases occur. Does every triple occur eventually? - T. D. Noe, Aug 17 2012

See A238893 for the three bases. By far, the most common bases are (2,4,8). - T. D. Noe, Mar 07 2014 (exception are in A260184. - Giovanni Resta and Robert G. Wilson v, Jul 17 2015).

			10 is palindromic in bases 3, 4, and 9.
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. - _Giovanni Resta_ and _Robert G. Wilson v_, Jul 17 2015

T. D. Noe, Table of n, a(n) for n = 1..1234
Rick Regan, Finding numbers that are palindromic in multiple bases
Index entries for sequences related to palindromes

Cf. A050813, A214423, A214424, A214426 (palindromic in 0-2 and 4 bases).

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 3, AppendTo[t, n]]]; t

A050812(n) = 3.

The intersection of A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955 & A002113 which yields just three members. - Giovanni Resta and Robert G. Wilson v, Jul 17 2015

A331891 Negabinary palindromes: nonnegative numbers whose negabinary expansion (A039724) is palindromic.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 17, 21, 23, 31, 43, 51, 57, 65, 77, 85, 87, 103, 127, 143, 155, 171, 195, 211, 217, 233, 257, 273, 285, 301, 325, 341, 343, 375, 423, 455, 479, 511, 559, 591, 603, 635, 683, 715, 739, 771, 819, 851, 857, 889, 937, 969, 993, 1025, 1073, 1105, 1117

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 its negabinary representation is 101 which is palindromic.

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

Cf. A002113, A006995, A014190, A039724, A094202, A331191.

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]]; Select[Range[0, 1200], PalindromeQ @ negabin[#] &]

A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 13, 20, 30, 35, 40, 44, 49, 71, 88, 102, 119, 170, 182, 194, 204, 216, 238, 242, 254, 266, 276, 288, 409, 450, 484, 525, 559, 580, 621, 655, 696, 986, 1015, 1044, 1068, 1097, 1150, 1160, 1189, 1218, 1242, 1271, 1334, 1363, 1392, 1396, 1425, 1454

Offset: 1

Amiram Eldar, Mar 12 2022

A052937(n) = A000129(n+1)+1 is a term for n>0, since its minimal Pell representation is 10...01 with n-1 0's between two 1's.
A048739 is a subsequence since these are repunit numbers in the minimal Pell representation.
A001109 is a subsequence. The minimal Pell representation of A001109(n), for n>1, is 1010...01, with n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A317204(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3             11
   4     6            101
   5     8            111
   6    13           1001
   7    20           1111
   8    30          10001
   9    35          10101
  10    40          10201

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

Cf. A000129, A317204.
Subsequences: A001109, A048739, A052937 \ {2}.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; PalindromeQ[IntegerDigits[Total[3^(s - 1)], 3]]]; Select[Range[0, 1500], q]

A352341 Numbers whose maximal Pell representation (A352339) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 10, 20, 27, 40, 49, 54, 58, 63, 68, 88, 93, 119, 136, 150, 167, 221, 238, 288, 300, 310, 322, 334, 338, 360, 372, 382, 394, 406, 508, 530, 542, 696, 737, 771, 812, 833, 867, 908, 942, 983, 1242, 1276, 1317, 1392, 1681, 1710, 1734, 1763, 1792, 1802

Offset: 1

Amiram Eldar, Mar 12 2022

A000129(n) - 2 is a term for n > 1. The maximal Pell representations of these numbers are 0, 11, 121, 1221, 12221, ... (0 and A132583).
A048739 is a subsequence since these are the repunit numbers in the maximal Pell representation.
A065113 is a subsequence since the maximal Pell representation of A065113(n) is 2*n 2's.

			The first 10 terms are:
   n  a(n)  A352339(a(n))
  --  ----  -------------
   1    0               0
   2    1               1
   3    3              11
   4    6              22
   5    8             111
   6   10             121
   7   20            1111
   8   27            1221
   9   40            2222
  10   49           11111

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

Cf. A000129, A048739, A065113, A132583, A352339.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazy[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Select[Range[0, 2000], PalindromeQ[lazy[#]] &]

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

cp's OEIS Frontend

A351717 Numbers whose maximal (or lazy) Lucas representation (A130311) is palindromic.

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A029971 Palindromic primes in base 3.

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A352087 Numbers whose minimal (or greedy) tribonacci representation (A278038) is palindromic.

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A352105 Numbers whose maximal tribonacci representation (A352103) is palindromic.

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A029958 Numbers that are palindromic in base 13.

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A214425 Numbers n palindromic in exactly three bases b, 2 <= b <= 10.

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A331891 Negabinary palindromes: nonnegative numbers whose negabinary expansion (A039724) is palindromic.

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A352319 Numbers whose minimal (or greedy) Pell representation (A317204) is palindromic.

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