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.

Showing 1-10 of 21 results. Next

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

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

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)

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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

Formula

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)

Extensions

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

Views

Author

Robert G. Wilson v, Sep 30 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Robert G. Wilson v, Sep 30 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

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, 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 *)
  • Python
    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

Formula

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

Views

Author

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

Keywords

Comments

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

Examples

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

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, 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 *)

Formula

Intersection of A014190 and A029953.

A259376 Palindromic numbers in bases 4 and 6 written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 21, 55, 215, 819, 1885, 7373, 7517, 12691, 14539, 69313, 196606, 1856845, 3314083, 5494725, 33348861, 223892055, 231755895, 322509617, 3614009815, 4036503055, 4165108015, 9233901154, 9330794722, 12982275395, 107074105033, 186398221946, 270747359295, 401478741365, 1809863435625, 2281658774290, 11931403417210, 12761538567790, 12887266632430, 15822654274715, 30255762326713, 46164680151002, 323292550693473, 329536806222753
Offset: 1

Views

Author

Eric A. Schmidt and Robert G. Wilson v, Jul 15 2015

Keywords

Examples

			55 is in the sequence because 55_10 = 131_6 = 313_4.

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, 6]; If[palQ[pp, 4], AppendTo[lst, pp]; Print[pp]]; k++]; lst

Formula

Intersection of A014190 and A029953.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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.

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, 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 *)

Formula

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

Views

Author

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

Keywords

Examples

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

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, 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 *)

Formula

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

Views

Author

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

Keywords

Examples

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

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, 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 *)

Formula

Intersection of A014192 and A029803.

Extensions

Corrected and extended by Giovanni Resta, Jul 16 2015
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