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

1,2

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

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)

Sequence in context: A035061 A302403 A096738 * A117284 A137377 A274793

Adjacent sequences: A167854 A167855 A167856 * A167858 A167859 A167860

Keyword

nonn

Author

T. D. Noe, Nov 13 2009

Status

approved