A167857 Numbers whose divisors are represented by an integer polynomial.

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

Offset: 1

Author

T. D. Noe, Nov 13 2009

Keywords

• nonn

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?

Examples

			The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A108164, A108166, A112774 (forms of semiprimes)

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)

Programs

Mathematica

Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]

PARI

is(n)=my(d=divisors(n));denominator(content(polinterpolate([0..#d-1],d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016