A060878 -id:A060878 - cp's OEIS frontend

A060879 Intrinsic 9-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

257, 273, 297, 313, 325, 341, 365, 381, 387, 403, 427, 443, 455, 471, 495, 511, 6562, 6643, 6724, 6832, 6913, 6994, 7102, 7183, 7264, 7300, 7381, 7462, 7570, 7651, 7732, 7840, 7921, 8002, 8038, 8119, 8200, 8308, 8389, 8470, 8578, 8659

Offset: 1

Harvey P. Dale, May 05 2001

All numbers are intrinsic 1- and (for n>2) 2-palindromes, almost all numbers are intrinsic 3-palindromes and very few numbers are intrinsic k-palindromes for k >= 4. - Franklin T. Adams-Watters, Jul 29 2011

Peter Kagey, Table of n, a(n) for n = 1..10000
A. J. Di Scala and M. Sombra, Intrinsic Palindromic Numbers, arXiv:math/0105022 [math.GM], 2001.
A. J. Di Scala and M. Sombra, Intrinsic Palindromes, Fib. Quart. 42, no. 1, Feb. 2004, pp. 76-81.

Cf. A060873, A060874, A060875, A060876, A060877, A060878, A060947, A060948, A060949.

A060875 Intrinsic 5-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

Original entry on oeis.org

17, 21, 27, 31, 82, 91, 100, 112, 121, 130, 142, 151, 160, 164, 173, 182, 194, 203, 212, 224, 233, 242, 257, 273, 289, 305, 325, 341, 357, 373, 393, 409, 425, 441, 461, 477, 493, 509, 514, 530, 546, 562, 582, 598, 614, 626, 630, 650, 651, 666

Offset: 1

Harvey P. Dale, May 05 2001

All numbers are intrinsic 1- and 2-palindromes, almost all numbers are intrinsic 3-palindromes and very few numbers are intrinsic k-palindromes for k >= 4.

Peter Kagey, Table of n, a(n) for n = 1..10000
A. J. Di Scala and M. Sombra, Intrinsic Palindromic Numbers, arXiv:math/0105022 [math.GM], 2001.
A. J. Di Scala and M. Sombra, Intrinsic Palindromes, Fib. Quart. 42, no. 1, Feb. 2004, pp. 76-81.

Cf. A060873, A060874, A060876, A060877, A060878, A060879, A060947, A060948, A060949.

A060876 Intrinsic 6-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

Original entry on oeis.org

33, 45, 51, 63, 244, 280, 316, 328, 364, 400, 412, 448, 484, 488, 524, 560, 572, 608, 644, 656, 692, 728, 1025, 1105, 1185, 1265, 1285, 1365, 1445, 1525, 1545, 1625, 1705, 1785, 1805, 1885, 1965, 2045, 2050, 2130, 2210, 2290, 2310, 2390

Offset: 1

Harvey P. Dale, May 05 2001

All numbers are intrinsic 1- and 2-palindromes, almost all numbers are intrinsic 3-palindromes and very few numbers are intrinsic k-palindromes for k >= 4.

Peter Kagey, Table of n, a(n) for n = 1..10000
A. J. Di Scala and M. Sombra, Intrinsic Palindromic Numbers, arXiv:math/0105022 [math.GM], 2001.
A. J. Di Scala and M. Sombra, Intrinsic Palindromes, Fib. Quart. 42, no. 1, Feb. 2004, pp. 76-81.

Cf. A060873, A060874, A060875, A060877, A060878, A060879, A060947, A060948, A060949.

A060877 Intrinsic 7-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

Original entry on oeis.org

65, 73, 85, 93, 99, 107, 119, 127, 730, 757, 784, 820, 847, 874, 910, 937, 964, 976, 1003, 1030, 1066, 1093, 1120, 1156, 1183, 1210, 1222, 1249, 1276, 1312, 1339, 1366, 1402, 1429, 1456, 1460, 1487, 1514, 1550, 1577, 1604, 1640, 1667, 1694

Offset: 1

Harvey P. Dale, May 05 2001

All numbers are intrinsic 1- and 2-palindromes, almost all numbers are intrinsic 3-palindromes and very few numbers are intrinsic k-palindromes for k >= 4.

Peter Kagey, Table of n, a(n) for n = 1..10000
A. J. Di Scala and M. Sombra, Intrinsic Palindromic Numbers, arXiv:math/0105022 [math.GM], 2001.
A. J. Di Scala and M. Sombra, Intrinsic Palindromes, Fib. Quart. 42, no. 1, Feb. 2004, pp. 76-81.

Cf. A060873, A060874, A060875, A060876, A060878, A060879, A060947, A060947, A060949.

A100563 Number of bases less than sqrt(n) in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 0, 1, 0, 1, 2, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 3, 0, 2, 1, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 0, 2, 1, 2, 1, 0, 3, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 1, 2, 1, 3, 1, 0, 0, 2, 2

Offset: 1

Gordon Hamilton, Nov 29 2004

Is there a number m such that a(n) > 0 for all n > m? I call the set of numbers for which a(n)=0 "unkempt" for refusing to use a mirror in any base. Is there an infinite number of unkempt numbers? a(n) can be arbitrarily large.
The sequence A123586 gives the values of n where a(n)=0. - Robert G. Wilson v, Nov 01 2014
Is there a closed-form formula for this function? - Robert G. Wilson v, Nov 01 2014
From Robert G. Wilson v, Nov 26 2014: (Start)
The first occurrence, beginning at 0, of n is: 1, 5, 17, 65, 121, 562, 1432, 1477, 4369, 36582, 35101, 86677, 83161, 360361, 291721, 720721, 887041, 1496881, 1670761, 3931201, 3341521, 5654881, 7207201, 7761601,...
Positions where a(n)=k:
k = 0:  A123586;
k = 1:  5, 7, 9, 10, 13, 15, 16, 20, 23, 25, 27, 28, 29, 33, 34, 36, 37, 38, 40, ...;
k = 2:  17, 21, 26, 31, 46, 51, 52, 55, 57, 63, 67, 73, 78, 80, 82, 91, 92, 93, 98, ...;
k = 3:  65, 85, 100, 130, 154, 164, 170, 178, 191, 195, 203, 209, 242, 282, 292, ...;
k = 4:  121, 235, 255, 257, 273, 300, 325, 341, 343, 373, 400, 495, 601, 610, 626, 666, ...;
k = 5:  562, 676, 771, 819, 1009, 1111, 1220, 1333, 1365, 1441, 1543, 1978, 1981, 2000, ...;
k = 6:  1432, 2380, 2666, 2925, 3280, 4035, 4095, 4161, 4225, 4401, 4525, 4561, 4681, ...;
k = 7:  1477, 4097, 4591, 7141, 7993, 8191, 9640, 10081, 10297, 10626, 10858, 11761, ...; etc.
(End)

			100 is a palindrome in bases 3, 7 and 9, so a(100) = 3.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Wikipedia, Base 1

Cf. A060873, A060874, A060875, A060876, A060877, A060878, A060879, A060947, A060948, A060949, A123586.

Cf. A000042.

f[n_] := Module[{p}, Table[ p = IntegerDigits[n, b]; If[p == Reverse@ p, {b, p}, Sequence @@ {}], {b, 2, Sqrt@ n}]]; Array[ Length@ f@# &, 105] (* Robert G. Wilson v, Nov 01 2014 *)

a(n) = {my(nb = 0); for (b=2, sqrt(n), d = digits(n, b); nb+= (Vecrev(d) == d);); nb;} \\ Michel Marcus, Nov 05 2014

a(n) = A135551(n) - A033831(n). - Robert G. Wilson v, Nov 01 2014

a(58) from Robert G. Wilson v, Nov 05 2014

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A060879 Intrinsic 9-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

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A060875 Intrinsic 5-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

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A060876 Intrinsic 6-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

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A060877 Intrinsic 7-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

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A100563 Number of bases less than sqrt(n) in which n is a palindrome.

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