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A167857
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Numbers whose divisors are represented by an integer polynomial.
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1
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1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 22, 23, 25, 29, 31, 34, 37, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 67, 71, 73, 79, 82, 83, 85, 89, 91, 94, 97, 101, 103, 106, 107, 109, 113, 115, 118, 121, 127, 131, 133, 137, 139, 142, 145, 149, 151, 157, 163, 166, 167, 169, 171
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OFFSET
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1,2
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COMMENTS
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That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.
Every prime has this property, as do 1 and 9, the squares of primes of the form 6k+1, and semiprimes p*q with p and q both primes of the form 3k-1 or 3k+1. Terms of the form p^2*q also appear. We can find terms of the form p^m for any m. For example, 2311^13 is the smallest 13th power that appears. For any m, it seems that p^m appears for p a prime of the form k*m#+1, where m# is the product of the primes up to m. Are there terms with three distinct prime divisors?
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LINKS
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EXAMPLE
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The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.
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MATHEMATICA
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Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]
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PROG
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(PARI) is(n)=my(d=divisors(n)); denominator(content(polinterpolate([0..#d-1], d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016
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CROSSREFS
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Cf. A002476 (primes of the form 6k+1)
Cf. A132230 (primes of the form 30k+1)
Cf. A073103 (primes of the form 210k+1)
Cf. A073917 (least prime of the form k*prime(n)#+1)
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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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