A097855 -id:A097855 - cp's OEIS frontend

A029965 Palindromic in bases 9 and 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 191, 282, 373, 464, 555, 646, 656, 6886, 25752, 27472, 42324, 50605, 626626, 1540451, 1713171, 1721271, 1828281, 1877781, 1885881, 2401042, 2434342, 2442442, 2450542, 3106013, 3114113, 3122213, 3163613

Offset: 1

Patrick De Geest

Robert G. Wilson v and Ray Chandler, Table of n, a(n) for n = 1..66 (terms < 10^18, first 52 terms from Robert G. Wilson v)
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 9], AppendTo[l, a]], {n, 10000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
pQ[n_,k_]:=Reverse[x=IntegerDigits[n,k]]==x; t={}; Do[If[pQ[n,10] && pQ[n,9],AppendTo[t,n]],{n,3.2*10^6}]; t (* Jayanta Basu, May 25 2013 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 9] &] (* Robert Price, Nov 09 2019 *)

A007632 Numbers that are palindromic in bases 2 and 10.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15351, 32223, 39993, 53235, 53835, 73737, 585585, 1758571, 1934391, 1979791, 3129213, 5071705, 5259525, 5841485, 13500531, 719848917, 910373019, 939474939, 1290880921, 7451111547

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Charlton Harrison found a new record binary-decimal palindrome: 11000101111000010101010110100001110100000100000101110000101101010101000011110100011_2 = 7475703079870789703075747_10 on Dec 01 2001. The binary string contains 83 digits! Since then he has added twenty more terms. - Robert G. Wilson v, Jul 03 2006

Intersection of A002113 and A006995. - Reinhard Zumkeller, Jan 22 2012, Feb 07 2010

M. R. Calandra, Integers which are palindromic in both decimal and binary notation, J. Rec. Math., 18 (No. 1, 1985-1986), 47.
S. Pilpel, Some More Double Palindromic Integers, J. Rec. Math., 18 (1985), 174-176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eshed Shaham, Table of n, a(n) for n = 1..175 (Terms 1..147 variously by Robert G. Wilson v, Charlton Harrison, Ilya Nikulshin, Andrey Astrelin)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014 (see p. 9).
M. R. Calandra, Integers which are palindromic in both decimal and binary notation, J. Rec. Math., 18 (No. 1, 1985-1986), 47. (Annotated scanned copy) [With scan of J. Rec. Math. 18.3 (1985), pp. 168-173]
Patrick De Geest, Palindromic numbers beyond base 10
Charlton Harrison, Binary/Decimal Palindromes
Project Euler, Problem 36: Double-base palindromes
Eshed Shaham, Finding Binary & Decimal Palindromes

For number of terms less than or equal to 10^n, see A120764.

Cf. A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

a007632 n = a007632_list !! (n-1)
a007632_list = filter ((== 1) . a178225) a002113_list
-- Reinhard Zumkeller, Jan 22 2012

[n: n in [0..2*10^7] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 2) eq Reverse(Intseq(n, 2))]; // Vincenzo Librandi, Dec 31 2015

N:= 12: # to get all terms <= 10^N
ispal2:= proc(n) local L; if n::even then return false fi;
  L:= convert(n,base,2); evalb(L=ListTools:-Reverse(L)) end proc:
rev10:= proc(n) local L; L:= convert(n,base,10); add(10^i*L[-i-1],i=0..nops(L)-1) end proc:
pals10:= proc(d) local x,y;
  if d::even then [seq(x*10^(d/2)+rev10(x),x=10^(d/2-1)..10^(d/2)-1)]
  else [seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+rev10(x), y=0..9), x=10^((d-1)/2-1)..10^((d-1)/2)-1)]
  fi
end proc:
0, 1, 3, 5, 7, 9, seq(op(select(ispal2,pals10(d))),d=2..N); # Robert Israel, Dec 31 2015

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[ idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]]] ]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]] ]] ]] ]]; palQ[n_Integer, base_Integer]:= Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 2], AppendTo[l, a]], {n, 1000000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
b1=2; b2=10; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 2 10^7}]; lst (* Vincenzo Librandi, Dec 31 2015 *)
Select[Range[0,10^5], PalindromeQ[#] && # == IntegerReverse[#, 2] &] (* Robert Price, Nov 09 2019 *)

isok(n) = my(d = digits(n), b=binary(n)); (d == Vecrev(d)) && (b == Vecrev(b)); \\ Michel Marcus, Dec 31 2015

from itertools import chain
A007632_list = sorted([n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**6)),(int(str(x)+str(x)[-2::-1]) for x in range(10**6))) if bin(n)[2:] == bin(n)[:1:-1]]) # Chai Wah Wu, Nov 23 2014

One more term from George Russell (ger(AT)tzi.de), Nov 20 2000
Two further terms from Harvey P. Dale, Mar 09 2001
Further terms from George Russell (ger(AT)tzi.de), Nov 02 2001

A029966 Palindromic in bases 10 and 11.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 232, 343, 454, 565, 676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854

Offset: 1

Patrick De Geest

The first 79 terms all have an odd number of decimal digits. Is there a term with an even number of decimal digits? - Robert Israel, Nov 23 2014

Ray Chandler and Robert G. Wilson v, Table of n, a(n) for n = 1..79, a(66)-a(76) from Ray Chandler, Oct 31 2014
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

[n: n in [0..5000000] | Intseq(n) eq Reverse(Intseq(n))and Intseq(n, 11) eq Reverse(Intseq(n, 11))]; // Vincenzo Librandi, Nov 23 2014

N:= 11: # to get all terms with up to N decimal digits
qpali:= proc(k, b) local L; L:= convert(k, base, b); if L = ListTools:-Reverse(L) then k else NULL fi end proc:
digrev:= proc(k,b) local L,n; L:= convert(k,base,b); n:= nops(L); add(L[i]*b^(n-i),i=1..n); end proc:
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(qpali(n*10^m + digrev(n,10),11), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(qpali(n*10^(m+1)+y*10^m+digrev(n,10),11), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Res;  # Robert Israel, Nov 23 2014

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 12], AppendTo[l, a]], {n, 100000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
b1=10; b2=11; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10000000}]; lst (* Vincenzo Librandi, Nov 23 2014 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 11] &] (* Robert Price, Nov 09 2019 *)

A007633 Palindromic in bases 3 and 10.

Original entry on oeis.org

0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, 29092, 48884, 74647, 75457, 76267, 92929, 93739, 848848, 1521251, 2985892, 4022204, 4219124, 4251524, 4287824, 5737375, 7875787, 7949497, 27711772, 83155138, 112969211, 123464321

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

J. Meeus, Multibasic palindromes, J. Rec. Math., 18 (No. 3, 1985-1986), 168-173.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Table of n, a(n) for n = 1..65 (terms 1..41 from Robert Israel, terms 42..63 from Robert G. Wilson v)
M. R. Calandra, Integers which are palindromic in both decimal and binary notation, J. Rec. Math., 18 (No. 1, 1985-1986), 47. (Annotated scanned copy) [With scan of J. Rec. Math. 18.3 (1985), pp. 168-173]
Patrick De Geest, Palindromic numbers in other bases.

Cf. A007632, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

ND:= 12;  # to get all terms with <= ND decimal digits
rev10:= proc(n) option remember;
  rev10(floor(n/10)) + (n mod 10)*10^ilog10(n)
end;
for i from 0 to 9 do rev10(i):= i od:
rev3:= proc(n) option remember;
  rev3(floor(n/3)) + (n mod 3)*3^ilog[3](n)
end;
for i from 0 to 2 do rev3(i):= i od:
pali3:= n -> rev3(n) = n;
count:= 1:
A[1]:= 0:
for d from 1 to ND do
  d1:= ceil(d/2);
  for x from 10^(d1-1) to 10^d1 - 1 do
    if d::even then y:= x*10^d1+rev10(x)
    else y:= x*10^(d1-1)+rev10(floor(x/10));
    fi;
    if pali3(y) then
       count:= count+1;
       A[count]:= y;
    fi
  od:
od:
seq(A[i],i=1..count); # Robert Israel, Apr 20 2014

Do[ a = IntegerDigits[n]; b = IntegerDigits[n, 3]; If[a == Reverse[a] && b == Reverse[b], Print[n] ], {n, 0, 10^9} ]
NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 4], AppendTo[l, a]], {n, 100000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
pal3Q[n_]:=Module[{idn3=IntegerDigits[n,3]},idn3==Reverse[idn3]]; Select[ Range[ 0,1235*10^5],PalindromeQ[#]&&pal3Q[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2019 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 3] &] (* Robert Price, Nov 09 2019 *)

from itertools import chain
from gmpy2 import digits
A007633_list = sorted([n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**6)),(int(str(x)+str(x)[-2::-1]) for x in range(10**6))) if digits(n,3) == digits(n,3)[::-1]]) # Chai Wah Wu, Nov 23 2014

A029961 Palindromic in bases 4 and 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 55, 373, 393, 666, 787, 939, 7997, 53235, 55255, 55655, 57675, 506605, 1801081, 2215122, 3826283, 3866683, 5051505, 5226225, 5259525, 5297925, 5614165, 5679765, 53822835, 623010326, 954656459, 51717171715

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..56
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 4], AppendTo[l, a]], {n, 1000000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 4] &] (* Robert Price, Nov 09 2019 *)

A029970 Numbers that are palindromic in bases 10 and 15.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 828, 858, 888, 919, 949, 979, 1551, 2772, 23632, 25552, 60106, 67576, 465564, 477774, 489984, 515515, 527725, 17577571, 26144162, 28300382, 39399393, 47999974, 69455496, 2118008112, 8050880508

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..72 (first 69 terms from Ray Chandler)
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029731, A097855, A099165.

[n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 15) eq Reverse(Intseq(n, 15))]; // Vincenzo Librandi, Nov 23 2014

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 15], AppendTo[l, a]], {n, 200000}]; l (* Robert G. Wilson v, Sep 03 2004 *)
b1=10; b2=15; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10000000}]; lst (* Vincenzo Librandi, Nov 23 2014 *)
Select[Range[0, 10^5], PalindromeQ[#] && # == IntegerReverse[#, 15] &] (* Robert Price, Nov 09 2019 *)

A029967 Palindromic in bases 12 and 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 181, 555, 616, 676, 737, 797, 1111, 8008, 35953, 43934, 88888, 646646, 3192913, 5641465, 8364638, 9963699, 133373331, 139979931, 293373392, 520020025, 713171317, 796212697, 1393223931

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..57
Patrick De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029968, A029969, A029970, A029731, A097855, A099165.

Select[Range[0, 10^5], PalindromeQ[#] && # == IntegerReverse[#, 12] &] (* Robert Price, Nov 09 2019 *)

from gmpy2 import digits
def palQ(n,b): # check if n is a palindrome in base b
    s = digits(n,b)
    return s == s[::-1]
def palQgen10(l): # generator of palindromes in base 10 of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(10**(x-1),10**x):
                s = str(y)
                yield int(s+s[-2::-1])
            for y in range(10**(x-1),10**x):
                s = str(y)
                yield int(s+s[::-1])
A029967_list = [n for n in palQgen10(5) if palQ(n,12)] # Chai Wah Wu, Dec 01 2014

A029968 Palindromic in bases 13 and 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 222, 313, 353, 444, 575, 666, 797, 1111, 6776, 8778, 24542, 25452, 26362, 56265, 311113, 2377732, 2713172, 2832382, 2906092, 8864688, 10122101, 13055031, 20244202, 20944902, 23177132, 23877832

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..76
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029969, A029970, A029731, A097855, A099165.

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 13], AppendTo[l, a]], {n, 100000}]; l (* Robert G. Wilson v, Sep 03 2004 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 13] &] (* Robert Price, Nov 09 2019 *)

from gmpy2 import digits
def palQ(n,b): # check if n is a palindrome in base b
    s = digits(n,b)
    return s == s[::-1]
def palQgen10(l): # generator of palindromes in base 10 of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(10**(x-1),10**x):
                s = str(y)
                yield int(s+s[-2::-1])
            for y in range(10**(x-1),10**x):
                s = str(y)
                yield int(s+s[::-1])
A029968_list = [n for n in palQgen10(9) if palQ(n,13)]
# Chai Wah Wu, Dec 01 2014

A029969 Numbers that are palindromic in bases 10 and 14.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 323, 464, 717, 858, 999, 1111, 39593, 59095, 420024, 546645, 9046409, 9578759, 9813189, 535505535, 564303465, 595121595, 5736116375, 6758008576, 10476867401, 11652825611, 14203330241

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..70 (first 63 terms from Ray Chandler)
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029970, A029731, A097855, A099165.

[n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 14) eq Reverse(Intseq(n, 14))]; // Vincenzo Librandi, Nov 23 2014

NextPalindrome[n_] := Block[{l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]] ]] FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[idn, Ceiling[l/2]], Reverse[ Take[idn, Floor[l/2]] ]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[idfhn], Drop[ Reverse[ IntegerDigits[idfhn]], Mod[l, 2]] ]]] ]]]; palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; l = {0}; a = 0; Do[a = NextPalindrome[a]; If[ palQ[a, 14], AppendTo[l, a]], {n, 300000}]; l (* Robert G. Wilson v, Sep 03 2004 *)
b1=10; b2=14; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 1000000}]; lst (* Vincenzo Librandi, Nov 23 2014 *)
palQ[n_]:=Module[{idn14=IntegerDigits[n,14]},n==IntegerReverse[n]&&idn14==Reverse[idn14]]; Select[Range[10^7],palQ] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Apr 23 2016 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 14] &] (* Robert Price, Nov 09 2019 *)

A029731 Palindromic in bases 10 and 16.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 353, 626, 787, 979, 1991, 3003, 39593, 41514, 90209, 94049, 96369, 98689, 333333, 512215, 666666, 749947, 845548, 1612161, 2485842, 5614165, 6487846, 9616169, 67433476, 90999909, 94355349, 94544549, 119919911

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..81
Patrick De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A097855, A099165.

N:= 9: # to get all terms with up to N decimal digits
qpali:= proc(k, b) local L; L:= convert(k, base, b); if L = ListTools:-Reverse(L) then k else NULL fi end proc:
digrev:= proc(k,b) local L,n; L:= convert(k,base,b); n:= nops(L); add(L[i]*b^(n-i),i=1..n); end proc:
Res:= $0..9:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(qpali(n*10^m + digrev(n,10),16), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(qpali(n*10^(m+1)+y*10^m+digrev(n,10),16), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Res; # Robert Israel, Nov 23 2014

A029731Q = PalindromeQ@# && IntegerReverse[#, 16] == # &; Select[Range[10^5], A029731Q] (* JungHwan Min, Mar 02 2017 *)
Select[Range[10^7], Times @@ Boole@ Map[# == Reverse@ # &, {IntegerDigits@ #, IntegerDigits[#, 16]}] > 0 &] (* Michael De Vlieger, Mar 03 2017 *)

def palQ16(n): # check if n is a palindrome in base 16
    s = hex(n)[2:]
    return s == s[::-1]
def palQgen10(l): # unordered generator of palindromes of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,10**l):
            s = str(x)
            yield int(s+s[-2::-1])
            yield int(s+s[::-1])
A029731_list = sorted([n for n in palQgen10(6) if palQ16(n)])
# Chai Wah Wu, Nov 25 2014

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