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A293900
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Number of permutations of the divisors of n that are greater than 1, in which consecutive elements are not coprime and no divisor d may occur later than any divisor e if e < d and A007947(e) = A007947(d). That is, any subset of divisors sharing the same squarefree part occur always in ascending order inside the permutation.
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3
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0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 9, 1, 2, 2, 1, 1, 9, 1, 9, 2, 2, 1, 40, 1, 2, 1, 9, 1, 348, 1, 1, 2, 2, 2, 110, 1, 2, 2, 40, 1, 348, 1, 9, 9, 2, 1, 175, 1, 9, 2, 9, 1, 40, 2, 40, 2, 2, 1, 138660, 1, 2, 9, 1, 2, 348, 1, 9, 2, 348, 1, 1127, 1, 2, 9, 9, 2, 348, 1, 175, 1, 2, 1, 138660, 2, 2, 2, 40, 1, 138660, 2, 9, 2, 2, 2, 756, 1, 9, 9, 110
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OFFSET
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1,6
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COMMENTS
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This is a more restricted variant of A163820, inspired by David A. Corneth's suggestion (personal e-mail) for optimizing its computation.
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LINKS
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FORMULA
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Iff n = p^k for some prime p and k >= 1 [that is, n is a term of A000961 > 1], then a(n) = 1.
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EXAMPLE
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The proper divisors of 12 are 2, 3, 4, 6, 12. a(12) = 9 because we can find nine permutations of them such that consecutive elements d and e are not coprime (that is, gcd(d,e) > 1) and where no divisor d is ever followed by divisor e such that A007947(d) = A007947(e) and e < d. These nine allowed permutations are (note that 2 must become before 4 and 6 must become before 12):
[2, 4, 6, 3, 12],
[2, 4, 6, 12, 3],
[2, 6, 3, 12, 4],
[2, 6, 4, 12, 3],
[3, 6, 2, 4, 12],
[3, 6, 2, 12, 4],
[3, 6, 12, 2, 4],
[6, 2, 4, 12, 3],
[6, 3, 12, 2, 4].
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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