A224220 a(n) = smallest number k with property that if the base-n expansion of k is reversed, the result is a nontrivial multiple of k.

32, 75, 8, 245, 12, 21, 16, 1089, 15, 1859, 21, 39, 28, 4335, 24, 6137, 24, 57, 40, 11109, 33, 115, 39, 45, 52, 22707, 35, 27869, 40, 93, 64, 55, 51, 47915, 57, 111, 76, 65559, 48, 75809, 56, 129, 88, 99405, 69, 329, 60, 119, 65, 143259, 72, 265, 63, 95, 112, 198417, 87, 219539

Offset: 3

Author

N. J. A. Sloane, Apr 01 2013

Keywords

• nonn
• base

Comments

In other words, k divides (reversal of k in base n), and (k-reversed)/k > 1.

The numbers are written in base 10.

Theorem: The length of k (in base n) is 2 iff n>=5 and n+1 is composite, otherwise 4.

Examples

			The numbers a(n) for n = 3, ..., 11 written in base n are 1012, 1023, 13, 1045, 15, 25, 17, 1089, 14.
For example, 1012 (base 3) = 32 (base 10), and 2101 (base 3) = 64 (base 10) = 2*32.
		

References

• N. J. A. Sloane, paper in preparation.
• See A214927 for further references and links.

Links

• Michel Marcus, Table of n, a(n) for n = 3..400
• T. J. Kaczynski, Note on a Problem of Alan Sutcliffe, Math. Mag., 41 (1968), 84-86.
• Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132. Also arXiv:math/0511366 [math.HO], 2005.
• Alan Sutcliffe, Integers That Are Multiplied When Their Digits Are Reversed, Mathematics Magazine, 39 (1966), 282-287.

Crossrefs

Cf. A214927, A006093, A072668, A152619.

Programs

Mathematica

Table[k = 2; While[Nand[IntegerQ@ #, # != 1] &[FromDigits[#, n]/k] &@ Reverse@ IntegerDigits[k, n], k++]; k, {n, 3, 60}] (* Michael De Vlieger, Feb 26 2017 *)

PARI

isok(k, n) = {my(rk = fromdigits(Vecrev(digits(k, n)), n)); !(rk % k) && (rk > k);}
a(n) = {my(k = 1); while (!isok(k, n), k++); k;} \\ Michel Marcus, Feb 26 2017

Formula

If n=3 or n>3 and n+1 is prime, a(n) = (n^2-1)(n+1) (cf. A152619).