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A179197
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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