A090061 Numbers k divisible by exactly two nontrivial permutations (rearrangements) of the digits of k, excluding all permutations that result in digit loss.

571428, 867132, 874125, 923076, 5179428, 5714028, 5714280, 5714820, 5719428, 5971428, 8524710, 8571042, 8671320, 8679132, 8741250, 8749125, 8914752, 8957142, 9230760, 9239076, 37451268, 41957028, 42195708, 42713568, 42915780, 42971580, 43157286, 43751286, 48713562, 51374268

Offset: 1

Author

Chuck Seggelin, Nov 21 2003

Keywords

• nonn
• base

Comments

Trivial permutations are identified as those where the permutation = k itself. Digit loss occurs when a permutation has 0 in the most significant position, which drops off, leaving a number with fewer digits. For example, when k is 3105, the permutation 0315 is excluded because 315 has fewer digits than 3105.

In the first million values of k, there is only one term that is divisible by three lossless nontrivial permutations. That term is 857142 which is divisible by 142857, 285714 and 428571. Note that 857142 is equal to floor((6/7)*10^6).

Examples

			a(4)=923076 is a term because 923076 is divisible by both 230769 and 307692, two nontrivial permutations of 923076 with the same number of digits.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..18986 (terms <= 10^10)
• David A. Corneth, PARI program
• Chuck Seggelin, Numbers Divisible by Digit Permutations.

Crossrefs

Cf. A090056, A090057.

Programs

PARI

\\ See Corneth link

Extensions

a(5)-a(25) from Donovan Johnson, Sep 16 2009

More terms from David A. Corneth, Jun 08 2025