A029965 -id:A029965 - cp's OEIS frontend

A259377 Palindromic numbers in bases 3 and 7 written in base 10.

0, 1, 2, 4, 8, 16, 40, 100, 121, 142, 164, 242, 328, 400, 1312, 8200, 9103, 14762, 54008, 76024, 108016, 112048, 233920, 532900, 639721, 741586, 2585488, 3316520, 11502842, 24919360, 35664908, 87001616, 184827640, 4346524576, 5642510512, 11641189600, 65304259157, 68095147754, 469837033600, 830172165614, 17136683996456, 21772277941544, 22666883572232, 45221839119556

Offset: 1

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

			142 is in the sequence because 142_10 = 262_7 = 12021_3.

Giovanni Resta, Table of n, a(n) for n = 1..72
Index entries for sequences related to palindromes

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

(* 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, 7]; If[palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=7; 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, Jul 17 2015 *)

Intersection of A014190 and A029954.

A259378 Palindromic numbers in bases 4 and 7 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 85, 150, 235, 257, 8802, 9958, 13655, 14811, 189806, 428585, 786435, 9262450, 31946605, 34179458, 387973685, 424623193, 430421657, 640680742, 742494286, 1692399385, 22182595205, 30592589645, 1103782149121, 1134972961921, 1871644872505, 2047644601565, 3205015384750, 3304611554563, 3628335729863, 4467627704385

Offset: 1

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

			85 is in the sequence because 85_10 = 151_7 = 1111_4.

Giovanni Resta, Table of n, a(n) for n = 1..48
Index entries for sequences related to palindromes

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.

(* 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, 7]; If[palQ[pp, 4], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=4; b2=7; 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, Jul 17 2015 *)

Intersection of A014192 and A029954.

A259382 Palindromic numbers in bases 4 and 8 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 63, 65, 105, 130, 170, 195, 235, 325, 341, 357, 373, 4095, 4097, 4161, 4225, 4289, 6697, 6761, 6825, 6889, 8194, 8258, 8322, 8386, 10794, 10858, 10922, 10986, 12291, 12355, 12419, 12483, 14891, 14955, 15019, 15083, 20485, 20805, 21525, 21845

Offset: 1

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

			235 is in the sequence because 235_10 = 353_8 = 3223_4.

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

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.

(* 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, 4], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=4; 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 *)

Intersection of A014192 and A029803.

Corrected and extended by Giovanni Resta, Jul 16 2015

A259384 Palindromic numbers in bases 6 and 8 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 154, 178, 203, 5001, 7409, 315721, 567434, 1032507, 46823602, 56939099, 84572293, 119204743, 1420737297, 1830945641, 2115191225, 3286138051, 3292861699, 4061216947, 8094406311, 43253138565, 80375377033, 88574916241, 108218625313, 116606986537, 116755331881, 166787896538, 186431605610, 318743407660, 396619220597, 1756866976011, 4920262093249, 11760498311914, 15804478291811, 15813860880803, 24722285628901, 33004205249575, 55584258482529, 371039856325905, 401205063672537, 516268720555889

Offset: 1

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

			178 is in the sequence because 178_10 = 262_8 = 454_6.

Giovanni Resta, Table of n, a(n) for n = 1..74
Index entries for sequences related to palindromes

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.

(* 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, 6], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=6; b2=8; 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, Jul 17 2015 *)

Intersection of A029953 and A029803.

A259385 Palindromic numbers in bases 2 and 9 written in base 10.

Original entry on oeis.org

0, 1, 3, 5, 7, 127, 255, 273, 455, 6643, 17057, 19433, 19929, 42405, 1245161, 1405397, 1786971, 2122113, 3519339, 4210945, 67472641, 90352181, 133638015, 134978817, 271114881, 6080408749, 11022828069, 24523959661, 25636651261, 25726334461, 28829406059, 1030890430479, 1032991588623, 1085079274815, 1616662113341

Offset: 1

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

			273 is in the sequence because 273_10 = 333_9 = 100010001_2.

Giovanni Resta, Table of n, a(n) for n = 1..44
A.H.M. Smeets, Scatterplot of log_2(number is palindromic in base 2 and base b) versus b, for b in {3,5,6,7,9,10}
Index entries for sequences related to palindromes

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.

(* 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, 9]; If[palQ[pp, 2], AppendTo[lst, pp]; Print[pp]]; k++]; lst

def nextpal(n, base): # m is the first palindrome successor of n in base base
    m, pl = n+1, 0
    while m > 0:
        m, pl = m//base, pl+1
    if n+1 == base**pl:
        pl = pl+1
    n = n//(base**(pl//2))+1
    m, n = n, n//(base**(pl%2))
    while n > 0:
        m, n = m*base+n%base, n//base
    return m
n, a2, a9 = 0, 0, 0
while n <= 30:
    if a2 < a9:
        a2 = nextpal(a2,2)
    elif a9 < a2:
        a9 = nextpal(a9, 9)
    else: # a2 == a9
        print(a2, end=",")
        a2, a9, n = nextpal(a2,2), nextpal(a9,9), n+1 # A.H.M. Smeets, Jun 03 2019

Intersection of A006995 and A029955.

A259386 Palindromic numbers in bases 3 and 9 written in base 10.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 20, 40, 80, 82, 91, 100, 164, 173, 182, 328, 364, 400, 656, 692, 728, 730, 820, 910, 1460, 1550, 1640, 2920, 3280, 3640, 5840, 6200, 6560, 6562, 6643, 6724, 7300, 7381, 7462, 8038, 8119, 8200, 13124, 13205, 13286, 13862, 13943, 14024, 14600, 14681, 14762, 26248, 26572, 26896, 29200, 29524, 29848, 32152, 32476, 32800, 52496, 52820, 53144, 55448, 55772, 56096, 58400, 58724, 59048, 59050, 59860, 60670, 65620, 66430, 67240, 72190, 73000, 73810

Offset: 1

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

			40 is in the sequence because 40_10 = 44_9 = 1111_3.

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

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.

(* 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, 9]; If[palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=9; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 80000}]; lst (* Vincenzo Librandi, Jul 17 2015 *)

Intersection of A014190 and A029955.

A259390 Palindromic numbers in bases 7 and 9 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 40, 50, 100, 164, 200, 264, 300, 328, 400, 2000, 3550, 8200, 10252, 14510, 14762, 22800, 45600, 164900, 201720, 400200, 532900, 555013, 738100, 2756120, 2913368, 3344352, 3501600, 4084000, 12990350, 22674550, 194062432, 1684866370, 2225211080, 13575144288, 15127811455, 20404027400, 20537111057, 22668403353, 30862471355, 83714515310, 84668107250, 796259955485, 1202029647736, 2088800185930, 20268849562000

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 17 2015

			264 is in the sequence because 264_10 = 323_9 = 525_7.

Giovanni Resta, Table of n, a(n) for n = 1..86
Index entries for sequences related to palindromes

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.

(* 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, 9]; If[palQ[pp, 7], AppendTo[lst, pp]; Print[pp]]; k++]; lst

Intersection of A029954 and A029955.

A118594 Palindromes in base 3 (written in base 3).

Original entry on oeis.org

0, 1, 2, 11, 22, 101, 111, 121, 202, 212, 222, 1001, 1111, 1221, 2002, 2112, 2222, 10001, 10101, 10201, 11011, 11111, 11211, 12021, 12121, 12221, 20002, 20102, 20202, 21012, 21112, 21212, 22022, 22122, 22222, 100001, 101101, 102201, 110011, 111111, 112211, 120021

Offset: 1

Martin Renner, May 08 2006

The number of n-digit terms is given by A225367. - M. F. Hasler, May 05 2013 [Moved here on May 08 2013]
Digit-wise application of A000578 (and also superposition of a(n) with its horizontal OR vertical reflection) yields A006072. - M. F. Hasler, May 08 2013
Equivalently, palindromes k (written in base 10) such that 4*k is a palindrome. - Bruno Berselli, Sep 12 2018

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A007089, A014190, A057148, A118595, A118596, A118597, A118598, A118599, A118600, A002113.

(* get NextPalindrome from A029965 *) Select[NestList[NextPalindrome, 0, 1110], Max@IntegerDigits@# < 3 &] (* Robert G. Wilson v, May 09 2006 *)
Select[FromDigits/@Tuples[{0,1,2},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

{for(l=1,5,u=vector((l+1)\2,i,10^(i-1)+(2*i-11&&i==1,2]), print1(v*u",")))} \\ The n-th term could be produced by using (partial sums of) A225367 to skip all shorter terms, and then skipping the adequate number of vectors v until n is reached.  - M. F. Hasler, May 08 2013

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1, 2]
    for d in count(2):
        for start in "12":
            for rest in product("012", repeat=d//2-1):
                left = start + "".join(rest)
                for mid in [[""], ["0", "1", "2"]][d%2]:
                    yield int(left + mid + left[::-1])
print(list(islice(agen(), 42))) # Michael S. Branicky, Mar 29 2022

from sympy import integer_log
from gmpy2 import digits
def A118594(n):
    if n == 1: return 0
    y = 3*(x:=3**integer_log(n>>1,3)[0])
    return int((s:=digits(n-x,3))+s[-2::-1] if nChai Wah Wu, Jun 14 2024

[int(n.str(base=3)) for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

More terms from Robert G. Wilson v, May 09 2006

a(40) and beyond from Michael S. Branicky, Mar 29 2022

A171775 a(n) = smallest number M such that there exist bases b_2, b_3, ..., b_n with the property that M written in base b_k is a k-digit palindrome for all k=2..n.

Original entry on oeis.org

1, 3, 5, 52, 130, 1885, 1073741824, 4398046511104

Offset: 1

James G. Merickel, Dec 18 2009

a(n) is no more than 2^[(n-1)*(n-2)] for n > 6 (and equals it for n = 7 and 8 at least). The reason for this bound is that for this number for each length from n down to 3 there is at least one power of 2, 2^k, such that in base b = 2^k-1 the binomial expansion of (b+1)^floor([(n-1)*(n-2)]/k) multiplied by the remaining small power of 2 gives a palindromic expression not requiring carries in base b. James G. Merickel, Aug 05 2015

			a(6)=1885: the bases are 1884 (1885 is 11 in base 1884), 14 (1885 is 989 in base 14), 12 (it is 1111 in base 12), 6 (it is 12421 in base 6), and 4 (it is 131131 in base 4).

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029804, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165, A087155, A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A253294.

a(7) and a(8) added by James G. Merickel, Feb 04 2010
Offset changed to 1, with corresponding addition of a(1) by James G. Merickel, Jul 24 2015
Comment corrected and explained.James G. Merickel, Aug 05 2015
Definition and example rewritten by N. J. A. Sloane, Aug 05 2015

A259381 Palindromic numbers in bases 3 and 8 written in base 10.

Original entry on oeis.org

0, 1, 2, 4, 121, 130, 203, 316, 8578, 9490, 17492, 944035, 1141652, 1276916, 1554173, 58961443, 67470916, 4099065139, 5691134677, 81452592329, 81473867465, 419572845958, 21056462595764, 363376288168081

Offset: 1

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

			121 is in the sequence because 121_10 = 171_8 = 11111_3.

Giovanni Resta, Table of n, a(n) for n = 1..43
Index entries for sequences related to palindromes

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.

(* 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, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=8; 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, Jul 17 2015 *)

Intersection of A014190 and A029803.

cp's OEIS Frontend

A259377 Palindromic numbers in bases 3 and 7 written in base 10.

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A259378 Palindromic numbers in bases 4 and 7 written in base 10.

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A259382 Palindromic numbers in bases 4 and 8 written in base 10.

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A259384 Palindromic numbers in bases 6 and 8 written in base 10.

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A259385 Palindromic numbers in bases 2 and 9 written in base 10.

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A259386 Palindromic numbers in bases 3 and 9 written in base 10.

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A259390 Palindromic numbers in bases 7 and 9 written in base 10.

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A118594 Palindromes in base 3 (written in base 3).

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