A029966 -id:A029966 - cp's OEIS frontend

A029731 Palindromic in bases 10 and 16.

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

A029962 Palindromic in bases 5 and 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 88, 252, 282, 626, 676, 1221, 15751, 18881, 10088001, 10400401, 27711772, 30322303, 47633674, 65977956, 808656808, 831333138, 831868138, 836131638, 836181638, 2512882152, 2596886952, 2893553982, 6761551676

Offset: 1

Patrick De Geest

Robert G. Wilson v, Table of n, a(n) for n = 1..79
P. De Geest, Palindromic numbers beyond base 10

Cf. A007632, A007633, A029961, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970.

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, 5], AppendTo[l, a]], {n, 200000}]; l (* Robert G. Wilson v, Sep 30 2004 *)
Select[Range[0, 10^5],
PalindromeQ[#] && # == IntegerReverse[#, 5] &] (* Robert Price, Nov 09 2019 *)

A029804 Numbers that are palindromic in bases 8 and 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 121, 292, 333, 373, 414, 585, 3663, 8778, 13131, 13331, 26462, 26662, 30103, 30303, 207702, 628826, 660066, 1496941, 1935391, 1970791, 4198914, 55366355, 130535031, 532898235, 719848917, 799535997, 1820330281

Offset: 1

Patrick De Geest

Intersection of A002113 and A029803. - Michel Marcus, Nov 20 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..75
Patrick De Geest, Palindromic numbers beyond base 10
Rick Regan, Code to generate this sequence

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

[n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 8) eq Reverse(Intseq(n, 8))]; // Vincenzo Librandi, Nov 23 2014

b1=8; b2=10; lst={}; Do[d1=IntegerDigits[n, b1];d2=IntegerDigits[n,b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 100000}]; lst (* Vincenzo Librandi, Nov 13 2014 *)
Select[Range[0,1820331000],PalindromeQ[#]&&IntegerDigits[#,8] == Reverse[ IntegerDigits[#,8]]&] (* Harvey P. Dale, Mar 18 2019 *)

isok(n) = (n==0) || ((d10=digits(n, 10)) && (d10==Vecrev(d10)) && (d8=digits(n, 8)) && (d8==Vecrev(d8))); \\ Michel Marcus, Nov 13 2014

ispal(n,r) = my(d=digits(n,r)); d==Vecrev(d);
for(n=0,10^7,if(ispal(n,10)&&ispal(n,8),print1(n,", "))); \\ Joerg Arndt, Nov 22 2014

def palQ8(n): # check if n is a palindrome in base 8
    s = oct(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])
A029804_list = sorted([n for n in palQgen10(6) if palQ8(n)])
# Chai Wah Wu, Nov 25 2014

More terms from Robert G. Wilson v, Sep 30 2004
Incorrect Mathematica program deleted by N. J. A. Sloane, Sep 01 2009
Terms 33 through 36 corrected by Rick Regan (exploringbinary(AT)gmail.com), Sep 01 2009

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

Cino Hilliard, Aug 31 2004

Robert G. Wilson v, Table of n, a(n) for n = 1..70 (first 67 terms from Ray Chandler)

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

[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

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 *)

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

Robert G. Wilson v, Sep 30 2004

Ray Chandler and Robert G. Wilson v, Table of n, a(n) for n = 1..115, terms a(88)-a(111) from Ray Chandler.

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

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 *)

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

A048268 Smallest palindrome greater than n in bases n and n+1.

Original entry on oeis.org

6643, 10, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 2

Ulrich Schimke (ulrschimke(AT)aol.com)

From A.H.M. Smeets, Jun 19 2019: (Start)
In the following, dig(expr) stands for the digit that represents the value of expression expr, and . stands for concatenation.
As for the naming of this sequence, the trivial 1 digit palindromes 0..dig(n-1) are excluded.
If a number m is palindromic in bases n and n+1, then m has an odd number of digits when represented in base n.
All three digit numbers in base n, that are palindromic in bases n and n+1 are given by:
101_3                         22_4                      for n = 3,
232_n                         1.dig(n).1_(n+1)
343_n                         2.dig(n-1).2_(n+1)
up to and including
dig(n-2).dig(n-1).dig(n-2)n  dig(n-3).4.dig(n-3)(n+1) for n > 3, and
dig(n-1).0.dig(n-1)n         dig(n-3).5.dig(n-3)(n+1) for n > 4.
Let d_L(n) be the number of integers with L digits in base n (L being odd), being palindromic in bases n and n+1, then:
d_1(n) = n for n >= 2 (see above),
d_3(n) = n-2 for n >= 5 (see above),
d_5(n) = n-1 for n >= 7 and n == 1 (mod 3),
d_5(n) = n-4 for n >= 7 and n in {0, 2} (mod 3), and
it seems that d_7(n) is of order O(n^2*log(n)) for n large enough. (End)

			a(14) = 2*14^2 + 3*14 + 2 = 436, which is 232_14 and 1e1_15.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A029965, A029966, A048269, A060792, A097928, A097929, A097930, A097931, A099145, A099146, A099147, A099153.

Do[ k = n + 2; While[ RealDigits[ k, n + 1 ][ [ 1 ] ] != Reverse[ RealDigits[ k, n + 1 ][ [ 1 ] ] ] || RealDigits[ k, n ][ [ 1 ] ] != Reverse[ RealDigits[ k, n ][ [ 1 ] ] ], k++ ]; Print[ k ], {n, 2, 75} ]
palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; f[n_] := Block[{k = n + 2}, While[ !palQ[k, n] || !palQ[k, n + 1], k++ ]; k]; Table[ f[n], {n, 2, 48}] (* Robert G. Wilson v, Sep 29 2004 *)

isok(j, n) = my(da=digits(j,n), db=digits(j,n+1)); (Vecrev(da)==da) && (Vecrev(db)==db);
a(n) = {my(j = n); while(! isok(j, n), j++); j;} \\ Michel Marcus, Nov 16 2017

Vec(x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Jun 30 2019

a(n) = 2n^2 + 3n + 2 for n >= 4 (which is 232_n and 1n1_(n+1)).
a(n) = A130883(n+1) for n > 3. - Robert G. Wilson v, Oct 08 2014
From Colin Barker, Jun 30 2019: (Start)
G.f.: x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)

More terms from Robert G. Wilson v, Aug 14 2000

A171706 The first n-fold intrinsically 7-palindromic number (represented in base 10).

Original entry on oeis.org

65, 186621, 3360633

Offset: 1

James G. Merickel, Dec 16 2009

Though this sequence is apt to never have many known members, its third is quite a nice coincidence for consideration.
A029965 runs just short of space to show it, and A053780 is a totally unrelated collection of palindromes that also includes 33633 in addition to 3360633.
a(4) > 10^17. - Hiroaki Yamanouchi, Aug 22 2015

			a(2)=186621 is 3555553 in base 6 and 1405041 in base 7.
a(3)=3360633 is 6281826 in base 9 and 1995991 in base 11.

Cf. A171701, A171702, A171703, A171704, A171705, A171741, A029966, A029965, A053780.

Typo in third term corrected by James G. Merickel, Dec 18 2009

A171741 The first 3-fold intrinsically n-palindromic number (given in base ten).

Original entry on oeis.org

18, 154, 19040, 267140, 2853326516320, 3360633

Offset: 2

James G. Merickel, Dec 17 2009

The large value for the 6-palindrome case, a(6)=2853326516320, has 139 as the smallest of the three bases.
Coincidentally, the following much shorter term, a(7)= 3360633, involves base 9 through 11, the conjunction of base-9 and base-10 palindromes (A029965) comes up a term short (assuming space limitations remain as they are at the beginning of 2011), and A053740 is a totally unrelated collection of palindromes that also contains 33633.
a(8) > 10^18. - Hiroaki Yamanouchi, Aug 22 2015

			a(5)=267140 is 6D4D6 in base 14, 54245 in base 15, and 29E92 in base 18.
a(7)=3360633 is 6281826 in base 9 and 1995991 in base 11.

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171742, A053780, A029966, A029965.

A099145 Numbers in base 10 that are palindromic in bases 7 and 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 121, 178, 235, 292, 300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, 450985, 502457, 626778, 786435, 10453500, 27924649

Offset: 1

Robert G. Wilson v, Sep 30 2004

Intersection of A029954 and A029803. - Michel Marcus, Oct 09 2014

			178 is in the sequence because 178_10 = 343_7 = 262_8.

Giovanni Resta, Table of n, a(n) for n = 1..48 (first 37 terms from Robert G. Wilson v)

Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099146, A029965, A029966, A099147, A099153.

palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 150000000], palQ[ #, 7] && palQ[ #, 8] &]

A099146 Numbers in base 10 that are palindromic in bases 8 and 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 154, 227, 300, 373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, 1257716, 1262900, 1268084, 1273268, 1294652, 1368461, 1373645, 1405397, 2067519, 63367795, 71877268, 98383349

Offset: 1

Robert G. Wilson v, Sep 30 2004

Intersection of A029803 and A029955. - Michel Marcus, Oct 09 2014

			227 is in the sequence because 227_10 = 343_8 = 272_9.

Giovanni Resta, Table of n, a(n) for n = 1..54 (first 41 terms from Robert G. Wilson v)

Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099145, A029965, A029966, A099147, A099153.

palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 250000000], palQ[ #, 8] && palQ[ #, 9] &]

Term 0 prepended by Robert G. Wilson v, Oct 08 2014

cp's OEIS Frontend

A029731 Palindromic in bases 10 and 16.

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Maple

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A029962 Palindromic in bases 5 and 10.

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A029804 Numbers that are palindromic in bases 8 and 10.

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Magma

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A097855 Numbers palindromic in bases 10 and 17.

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Magma

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A099165 Palindromic in bases 10 and 32.

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A048268 Smallest palindrome greater than n in bases n and n+1.

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Mathematica

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PARI

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A171706 The first n-fold intrinsically 7-palindromic number (represented in base 10).

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A171741 The first 3-fold intrinsically n-palindromic number (given in base ten).

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