A179197 Numbers k such that there exists no multiple of k whose digits are a permutation of all the digits of all the divisors of k.

3, 7, 9, 11, 12, 13, 17, 18, 19, 21, 22, 23, 27, 29, 31, 33, 36, 37, 39, 41, 43, 45, 47, 48, 49, 53, 54, 55, 57, 59, 61, 63, 67, 71, 72, 73, 74, 75, 77, 79, 81, 83, 84, 89, 91, 93, 97, 99, 101, 103, 107, 108, 109, 111, 113, 117, 121, 126, 129, 131, 135, 137, 139, 143, 144

Offset: 1

Author

Jon E. Schoenfield, Jul 02 2010

Keywords

• base
• nonn

Comments

Numbers k such that A077351(k)=0.

Let s(k) be the sum of the digits of all the divisors of k. The sequence must, of course, include every number k such that 3 divides k but does not divide s(k). Similarly, it must include every k such that 9 divides k but does not divide s(k). The sequence also includes many numbers with relatively few divisors, since the concatenation of their digits offers relatively few opportunities to obtain a multiple of k by permuting them. Of the sequence's 2544 terms below 10000, only four exist that (1) are not primes, (2) are not semiprimes, (3) are not prime powers, (4) are not numbers k that are divisible by 3 but having s(k) not divisible by 3, and (5) are not numbers k that are divisible by 9 but having s(k) not divisible by 9: 242, 2222, 5555, and 7777.

Examples

			The divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108, and concatenating all their digits gives the 19-digit number 1234691218273654108; no permutation of those 19 digits yields a result that is divisible by 108, so 108 is in the sequence.
The divisors of 14 are 1, 2, 7, and 14, and concatenating all their digits gives the 5-digit number 12714; those 5 digits can be permuted to yield a result (e.g., 21714) that is divisible by 14, so 14 is not in the sequence.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n=1..2544

Crossrefs

Cf. A077351.