cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    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

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Python
    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

A097855 Numbers palindromic in bases 10 and 17.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 252, 494, 545, 767, 818, 989, 2882, 4554, 61416, 94249, 177771, 256652, 335533, 1388831, 4165614, 8837388, 31744713, 102757201, 103595301, 123616321, 124454421, 207535702, 208373802, 212313212, 229232922
Offset: 1

Views

Author

Cino Hilliard, Aug 31 2004

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A099165.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 17) eq Reverse(Intseq(n, 17))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    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, 17], AppendTo[l, a]], {n, 40000}]; l (* Robert G. Wilson v, Sep 03 2004 *)
b1=10; b2=17; 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[#, 17] &] (* Robert Price, Nov 09 2019 *)

Extensions

More terms from Robert G. Wilson v, Sep 03 2004
Term 0 prepended by Robert G. Wilson v, Oct 07 2014

A099165 Palindromic in bases 10 and 32.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 66, 99, 363, 858, 1441, 2882, 5445, 6886, 9449, 15951, 19891, 21012, 29692, 32223, 54945, 369963, 477774, 564465, 585585, 609906, 672276, 717717, 780087, 804408, 912219, 1251521, 2639362, 3825283
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2004

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855.

Programs

  • Mathematica
    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, 32], AppendTo[l, a]], {n, 10000}]; l
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 32] &] (* Robert Price, Nov 09 2019 *)
  • Python
    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): # 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])
A099165_list = sorted([n for n in palQgen10(6) if palQ(n,32)])
# Chai Wah Wu, Nov 25 2014

A250408 Palindromic in bases 10 and 20.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 252, 6556, 6776, 7117, 10101, 12621, 20202, 22722, 30303, 1784871, 1786871, 1788871, 1913191, 1915191, 1917191, 1919191, 1444884441, 334495594433, 334843348433, 355110011553, 355746647553, 10614366341601, 14102600620141, 28095922959082, 38072044027083
Offset: 1

Views

Author

Robert G. Wilson v, Nov 22 2014

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250409, A250410, A250411, A099165, A250412.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 20) eq Reverse(Intseq(n, 20))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; genPal[n_] := Block[{id = IntegerDigits@ n, insert = {{}, {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; k = 1; lst = {0, 1, 2, 3,  4, 5, 6, 7, 8, 9}; While[k < 1000001, s = Select[ genPal[k], palQ[#, 20] &]; If[s != {}, AppendTo[lst, s]; Print@ s; lst = Sort@ Flatten@ lst]; k++]; lst
b1=10; b2=20; 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 *)
  • Python
    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): # 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])
A250408_list = sorted([n for n in palQgen10(6) if palQ(n,20)])
# Chai Wah Wu, Nov 25 2014

A250411 Palindromic in bases 10 and 27.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 252, 616, 757, 838, 919, 10301, 13031, 15951, 17871, 65856, 1197911, 2287822, 4385834, 5475745, 5549455, 6278726, 6639366, 7368637, 7573757, 8663668, 8737378, 9392939, 9466649, 9827289, 67166176, 214171412, 609808906, 836040638, 2132882312, 2487997842
Offset: 1

Views

Author

Robert G. Wilson v, Nov 23 2014

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250409, A250410, A099165, A250412.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 27) eq Reverse(Intseq(n, 27))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; genPal[n_] := Block[{id = IntegerDigits@ n, insert = {{}, {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; k = 1; lst = {0, 1, 2, 3,  4, 5, 6, 7, 8, 9}; While[k < 1000001, s = Select[ genPal[k], palQ[#, 27] &]; If[s != {}, AppendTo[lst, s]; Print@ s; lst = Sort@ Flatten@ lst]; k++]; lst
b1=10; b2=36; 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 *)

A250412 Palindromic in bases 10 and 36.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1221, 1441, 2882, 5115, 12321, 16861, 19491, 21112, 30803, 33433, 36063, 37973, 42224, 159951, 741147, 987789, 1301031, 1867681, 3315133, 4306034, 5182815, 5927295, 6918196, 6950596, 9242429, 9488849, 10066001, 48655684
Offset: 1

Views

Author

Robert G. Wilson v, Nov 23 2014

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250409, A250410, A250411, A099165.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 36) eq Reverse(Intseq(n, 36))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; genPal[n_] := Block[{id = IntegerDigits@ n, insert = {{}, {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; k = 1; lst = {0, 1, 2, 3,  4, 5, 6, 7, 8, 9}; While[k < 1000001, s = Select[ genPal[k], palQ[#, 36] &]; If[s != {}, AppendTo[lst, s]; Print@ s; lst = Sort@ Flatten@ lst]; k++]; lst
b1=10; b2=36; 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[0,49*10^6],PalindromeQ[#]&&IntegerDigits[#,36]== Reverse[ IntegerDigits[ #,36]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 04 2019 *)

A250409 Palindromic in bases 10 and 24.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 525, 575, 2332, 4664, 6226, 6996, 8558, 10001, 11011, 12621, 13631, 374473, 1851581, 2207022, 2267622, 2762672, 3118113, 3673763, 4029204, 4296924, 4925294, 5103015, 5836385, 6014106, 6281826, 6747476, 7132317, 7192917, 7658567, 7865687
Offset: 1

Views

Author

Robert G. Wilson v, Nov 22 2014

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250410, A250411, A099165, A250412.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 24) eq Reverse(Intseq(n, 24))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; genPal[n_] := Block[{id = IntegerDigits@ n, insert = {{}, {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; k = 1; lst = {0, 1, 2, 3,  4, 5, 6, 7, 8, 9}; While[k < 1000001, s = Select[ genPal[k], palQ[#, 24] &]; If[s != {}, AppendTo[lst, s]; Print@ s; lst = Sort@ Flatten@ lst]; k++]; lst
b1=10; b2=24; 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[0,79*10^5],PalindromeQ[#]&&IntegerDigits[#,24]== Reverse[ IntegerDigits[ #,24]]&] (* Harvey P. Dale, Mar 17 2023 *)

A250410 Numbers palindromic in bases 10 and 25.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 494, 626, 676, 1001, 6886, 7887, 8338, 9339, 622226, 626626, 2828282, 2859582, 3304033, 3309033, 3330333, 3335333, 3361633, 3366633, 3392933, 3397933, 6603066, 6608066, 6634366, 6639366, 8986898, 9400049, 9405049, 9431349, 9436349, 9462649, 9467649, 9493949, 9498949
Offset: 1

Views

Author

Robert G. Wilson v, Nov 22 2014

Keywords

Links

Crossrefs

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A250408, A250409, A250411, A099165, A250412.

Programs

  • Magma
    [n: n in [0..10000000] | Intseq(n) eq Reverse(Intseq(n))and Intseq(n, 25) eq Reverse(Intseq(n, 25))]; // Vincenzo Librandi, Nov 23 2014
  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; genPal[n_] := Block[{id = IntegerDigits@ n, insert = {{}, {0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; k = 1; lst = {0, 1, 2, 3,  4, 5, 6, 7, 8, 9}; While[k < 1000001, s = Select[ genPal[k], palQ[#, 25] &]; If[s != {}, AppendTo[lst, s]; Print@ s; lst = Sort@ Flatten@ lst]; k++]; lst
  • Python
    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): # 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])
A250410_list = sorted([n for n in palQgen10(6) if palQ(n,25)])
# Chai Wah Wu, Nov 25 2014

A259380 Palindromic numbers in bases 2 and 8 written in base 10.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 27, 45, 63, 65, 73, 195, 219, 325, 341, 365, 381, 455, 471, 495, 511, 513, 585, 1539, 1755, 2565, 2709, 2925, 3069, 3591, 3735, 3951, 4095, 4097, 4161, 4617, 4681, 12291, 12483, 13851, 14043, 20485, 20613, 20805, 20933, 21525, 21653, 21845, 21973, 23085, 23213, 23405, 23533, 24125, 24253, 24445, 24573, 28679, 28807, 28999, 29127, 29719, 29847
Offset: 1

Views

Author

Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015

Keywords

Examples

			2709 is in the sequence because 2709_10 = 5225_8 = 101010010101_2.

Links

Crossrefs

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376,
A097930, A182234, A259377, A259378, A249156, A097931, A259380, A259381, A259382, A259383,
A259384, A099145, A259385, A259386, A259387, A259388, A259389, A259390, A099146, A007632,
A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968,
A029969, A029970, A029731, A097855, A250408, A250409, A250410, A250411, A099165, A250412.

Programs

  • Mathematica
    (* first load nthPalindromeBase from A002113 *) palQ[n_Integer, base_Integer] := Block[{}, Reverse[ idn = IntegerDigits[n, base]] == idn]; k = 0; lst = {}; While[k < 21000000, pp = nthPalindromeBase[k, 8]; If[palQ[pp, 2], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=2; b2=8; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 30000}]; lst (* Vincenzo Librandi, Jul 17 2015 *)

Formula

Intersection of A006995 and A029803.
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