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

3,1

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.

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.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A214927, A006093, A072668, A152619.

Sequence in context: A110230 A250756 A303583 * A007797 A222546 A240054

Adjacent sequences: A224217 A224218 A224219 * A224221 A224222 A224223

Keyword

nonn,base

Author

N. J. A. Sloane, Apr 01 2013

Status

approved