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

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

Author

Gordon Hamilton, Nov 29 2004

Keywords

• easy
• base
• nonn

Comments

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)

Examples

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

Links

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

Crossrefs

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

Cf. A000042.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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