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A167857 Numbers whose divisors are represented by an integer polynomial. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 29 16:23 EDT 2022. Contains 357090 sequences. (Running on oeis4.)