A060792 -id:A060792 - cp's OEIS frontend

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

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

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

A249156 Palindromic in bases 5 and 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 24, 57, 78, 114, 342, 624, 856, 1432, 10308, 12654, 27616, 100056, 537856, 593836, 769621, 1434168, 1473368, 1636104, 1823544, 1862744, 17968646, 18108296, 22412057, 34713713, 34853363, 39280254, 159690408, 663706192

Offset: 1

Ray Chandler, Oct 27 2014

Intersection of A029952 and A029954.

			114 is a term since 114 = 424 base 5 and 114 = 222 base 7.

G. Resta, Table of n, a(n) for n = 1..72 (first 60 terms from Ray Chandler)
Attila Bérczes and Volker Ziegler, On Simultaneous Palindromes, arXiv:1403.0787 [math.NT], 2014.

Cf. A007632, A060792, A249155, A249157, A249158.

palQ[n_Integer,base_Integer]:=Block[{idn=IntegerDigits[n,base]},idn==Reverse[idn]];Select[Range[10^6]-1,palQ[#,5]&&palQ[#,7]&]

isok(n) = my(df = digits(n, 5), ds = digits(n, 7)); (Vecrev(df)==df) && (Vecrev(ds)==ds); \\ Michel Marcus, Oct 31 2017

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 palQgen(l,b): # unordered generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,b**l):
            s = digits(x,b)
            yield int(s+s[-2::-1],b)
            yield int(s+s[::-1],b)
A249156_list = sorted([n for n in palQgen(8,5) if palQ(n,7)]) # Chai Wah Wu, Nov 25 2014

A182232 Numbers that are palindromic in bases 2 and 5.

Original entry on oeis.org

0, 1, 3, 31, 93, 119, 2709, 38937, 520831, 682341, 340134981, 7609865031, 17935304097, 26777829859, 169179279801, 567897922593, 286118927218753, 2996750083037781, 4738749440161121, 6299497017331917, 8829547069230943

Offset: 1

Alex Ratushnyak, Apr 19 2012

Intersection of A006995 and A029952. - Michel Marcus, Oct 08 2014

			2709 base 2 = 101010010101 and 2709 base 5 = 41314.

Ray Chandler, Table of n, a(n) for n = 1..25

Cf. A060792, A097856.

b1 = 2; b2 = 5; lst = {}; Do[d1 = IntegerDigits[n, b1]; d2 = IntegerDigits[n, b2]; If[d1 == Reverse[d1] && d2 == Reverse[d2], AppendTo[lst, n]], {n, 1000000}]; lst (* T. D. Noe, Apr 19 2012 *)

Term 0 prepended by Robert G. Wilson v, Oct 08 2014
a(12)-a(16) from Robert G. Wilson v, Oct 11 2014
a(17)-a(21) and b-file from Ray Chandler, Oct 24 2014

A182233 Numbers that are palindromic in bases 2 and 6.

Original entry on oeis.org

0, 1, 3, 5, 7, 21, 129, 427, 693, 819, 3999, 4257, 4593, 28539, 66433, 85093, 148617, 151497, 153513, 180213, 425971, 1040319, 1093281, 1508381, 1632995, 1974031, 1986127, 30522135, 30643095, 208080483, 1894216583, 6662648163, 8632935681

Offset: 1

Alex Ratushnyak, Apr 19 2012

Intersection of A006995 and A029953. - Michel Marcus, Oct 09 2014

			85093 base 2 = 10100110001100101 and 85093 base 6 = 1453541.

Ray Chandler, Table of n, a(n) for n = 1..70 (terms < 10^18)

Cf. A006995 (base 2), A029953 (base 6).

Cf. A060792 (base 2 and 3), A097856 (base 2 and 4).

b1 = 2; b2 = 6; lst = {}; Do[d1 = IntegerDigits[n, b1]; d2 = IntegerDigits[n, b2]; If[d1 == Reverse[d1] && d2 == Reverse[d2], AppendTo[lst, n]], {n, 2000000}]; lst (* T. D. Noe, Apr 19 2012 *)

isok(n) = (d2=digits(n, 2)) && (d2==Vecrev(d2)) && (d6=digits(n, 6)) && (d6==Vecrev(d6)); \\ Michel Marcus, Oct 27 2014

a(28)-a(33) and b-file from Ray Chandler, Oct 27 2014

A182234 Numbers that are palindromic in bases 2 and 7.

Original entry on oeis.org

0, 1, 3, 5, 85, 107, 257, 5049, 9201, 11253, 11757, 210099, 399171, 512607, 786435, 12916899, 19992857, 22468309, 1052109663, 15935958711, 24051338445, 37344016593, 71859215265, 72822171105, 1566399158893, 3425211644643

Offset: 1

Alex Ratushnyak, Apr 19 2012

Intersection of A006995 and A029954. - Michel Marcus, Oct 09 2014

			786435 base 2 = 11000000000000000011 and 786435 base 7 = 6453546.

Ray Chandler, Table of n, a(n) for n = 1..31 (terms < 10^18)

Cf. A006995 (base 2), A029954 (base 7).

Cf. A060792 (base 2 and 3), A097856 (base 2 and 4).

b1 = 2; b2 = 7; lst = {}; Do[d1 = IntegerDigits[n, b1]; d2 = IntegerDigits[n, b2]; If[d1 == Reverse[d1] && d2 == Reverse[d2], AppendTo[lst, n]], {n, 1000000}]; lst (* T. D. Noe, Apr 19 2012 *)

a(19)-a(28) from Donovan Johnson, Apr 27 2012

b-file to 31 terms from Ray Chandler, Oct 27 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

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

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

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, 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 *)

Intersection of A006995 and A029803.

A259374 Palindromic numbers in bases 3 and 5 written in base 10.

Original entry on oeis.org

0, 1, 2, 4, 26, 52, 1066, 1667, 2188, 32152, 67834, 423176, 437576, 14752936, 26513692, 27711772, 33274388, 320785556, 1065805109, 9012701786, 9256436186, 12814126552, 18814619428, 201241053056, 478999841578, 670919564984, 18432110906024, 158312796835916, 278737550525722

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 14 2015

is only 0 regardless of the base,
is only 1 regardless of the base,
on the other hand is also 10 in base 2, denoted as 10_2,
is 3 in all bases greater than 3, but is 11_2 and 10_3.

			52 is in the sequence because 52_10 = 202_5 = 1221_3.

Giovanni Resta, Table of n, a(n) for n = 1..39
A.H.M. Smeets, Scatterplot of log_3(number is palindromic in base 3 and base b) versus b, for b in {2,4,5, 6,7,8,10}

Cf. A048268, A060792, A097856, A097928, A182232, A259374, A097929, A182233, A259375, A259376, A097930, A182234, A259377, A259378, A249156, A097931, A259380-A259384, A099145, A259385-A259390, A099146, A007632, A007633, A029961-A029964, A029804, A029965-A029970, A029731, A097855, A250408-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, 5]; If[ palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=5; 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 15 2015 *)

def nextpal(n,b): # returns the palindromic successor of n in base b
    m, pl = n+1, 0
    while m > 0:
        m, pl = m//b, pl+1
    if n+1 == b**pl:
        pl = pl+1
    n = (n//(b**(pl//2))+1)//(b**(pl%2))
    m = n
    while n > 0:
        m, n = m*b+n%b, n//b
    return m
n, a3, a5 = 0, 0, 0
while n <= 20000:
    if a3 < a5:
        a3 = nextpal(a3,3)
    elif a5 < a3:
        a5 = nextpal(a5,5)
    else: # a3 == a5
        print(n,a3)
        a3, a5, n = nextpal(a3,3), nextpal(a5,5), n+1
# A.H.M. Smeets, Jun 03 2019

Intersection of A014190 and A029952.

A259375 Palindromic numbers in bases 3 and 6 written in base 10.

Original entry on oeis.org

0, 1, 2, 4, 28, 80, 160, 203, 560, 644, 910, 34216, 34972, 74647, 87763, 122420, 221068, 225064, 6731644, 6877120, 6927700, 7723642, 8128762, 8271430, 77894071, 78526951, 539212009, 28476193256, 200267707484, 200316968444, 201509576804, 201669082004, 231852949304, 232018753064, 232039258376, 333349186006, 2947903946317, 5816975658914, 5817003372578, 11610051837124, 27950430282103, 81041908142188

Offset: 1

Eric A. Schmidt and Robert G. Wilson v, Jul 14 2015

Agrees with the number of minimal dominating sets of the halved cube graph Q_n/2 for at least n=1 to 5. - Eric W. Weisstein, Sep 06 2021

			28 is in the sequence because 28_10 = 44_6 = 1001_3.

Giovanni Resta, Table of n, a(n) for n = 1..64

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, 6]; If[palQ[pp, 3], AppendTo[lst, pp]; Print[pp]]; k++]; lst
b1=3; b2=6; 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 15 2015 *)

Intersection of A014190 and A029953.

cp's OEIS Frontend

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

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

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

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A249156 Palindromic in bases 5 and 7.

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A182232 Numbers that are palindromic in bases 2 and 5.

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A182233 Numbers that are palindromic in bases 2 and 6.

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A182234 Numbers that are palindromic in bases 2 and 7.

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