OFFSET
1,2
COMMENTS
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?
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.
MATHEMATICA
Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]
PROG
(PARI) is(n)=my(d=divisors(n)); denominator(content(polinterpolate([0..#d-1], d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 13 2009
STATUS
approved