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Search: a097929 -id:a097929
Displaying 1-10 of 21 results found. page 1 2 3
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A048268 Smallest palindrome greater than n in bases n and n+1. +10
27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
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)
LINKS
Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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)
EXAMPLE
a(14) = 2*14^2 + 3*14 + 2 = 436, which is 232_14 and 1e1_15.
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 *)
PROG
(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
CROSSREFS
Cf. A029965, A029966, A048269, A060792, A097928, A097929, A097930, A097931, A099145, A099146, A099147, A099153.
KEYWORD
nonn,easy,base
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
EXTENSIONS
More terms from Robert G. Wilson v, Aug 14 2000
STATUS
approved
A099145 Numbers in base 10 that are palindromic in bases 7 and 8. +10
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Intersection of A029954 and A029803. - Michel Marcus, Oct 09 2014
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..48 (first 37 terms from Robert G. Wilson v)
EXAMPLE
178 is in the sequence because 178_10 = 343_7 = 262_8.
MATHEMATICA
palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 150000000], palQ[ #, 7] && palQ[ #, 8] &]
CROSSREFS
Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099146, A029965, A029966, A099147, A099153.
KEYWORD
base,nonn
AUTHOR
Robert G. Wilson v, Sep 30 2004
STATUS
approved
A099146 Numbers in base 10 that are palindromic in bases 8 and 9. +10
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Intersection of A029803 and A029955. - Michel Marcus, Oct 09 2014
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..54 (first 41 terms from Robert G. Wilson v)
EXAMPLE
227 is in the sequence because 227_10 = 343_8 = 272_9.
MATHEMATICA
palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 250000000], palQ[ #, 8] && palQ[ #, 9] &]
CROSSREFS
Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099145, A029965, A029966, A099147, A099153.
KEYWORD
base,nonn
AUTHOR
Robert G. Wilson v, Sep 30 2004
EXTENSIONS
Term 0 prepended by Robert G. Wilson v, Oct 08 2014
STATUS
approved
A259380 Palindromic numbers in bases 2 and 8 written in base 10. +10
17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes
FORMULA
Intersection of A006995 and A029803.
EXAMPLE
2709 is in the sequence because 2709_10 = 5225_8 = 101010010101_2.
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 *)
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.
KEYWORD
base,nonn
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015
STATUS
approved
A259374 Palindromic numbers in bases 3 and 5 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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.
LINKS
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}
FORMULA
Intersection of A014190 and A029952.
EXAMPLE
52 is in the sequence because 52_10 = 202_5 = 1221_3.
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 *)
PROG
(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
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 14 2015
STATUS
approved
A259375 Palindromic numbers in bases 3 and 6 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..64
FORMULA
Intersection of A014190 and A029953.
EXAMPLE
28 is in the sequence because 28_10 = 44_6 = 1001_3.
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 *)
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 14 2015
STATUS
approved
A259376 Palindromic numbers in bases 4 and 6 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..64
FORMULA
Intersection of A014190 and A029953.
EXAMPLE
55 is in the sequence because 55_10 = 131_6 = 313_4.
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
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 15 2015
STATUS
approved
A259377 Palindromic numbers in bases 3 and 7 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..72
Index entries for sequences related to palindromes
FORMULA
Intersection of A014190 and A029954.
EXAMPLE
142 is in the sequence because 142_10 = 262_7 = 12021_3.
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 *)
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015
STATUS
approved
A259378 Palindromic numbers in bases 4 and 7 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..48
Index entries for sequences related to palindromes
FORMULA
Intersection of A014192 and A029954.
EXAMPLE
85 is in the sequence because 85_10 = 151_7 = 1111_4.
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 *)
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015
STATUS
approved
A259382 Palindromic numbers in bases 4 and 8 written in base 10. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes
FORMULA
Intersection of A014192 and A029803.
EXAMPLE
235 is in the sequence because 235_10 = 353_8 = 3223_4.
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 *)
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.
KEYWORD
nonn,base
AUTHOR
Eric A. Schmidt and Robert G. Wilson v, Jul 16 2015
EXTENSIONS
Corrected and extended by Giovanni Resta, Jul 16 2015
STATUS
approved
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