A167857 - OEIS

%I #5 Jan 29 2016 11:18:14

%S 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,

%T 58,59,61,67,71,73,79,82,83,85,89,91,94,97,101,103,106,107,109,113,

%U 115,118,121,127,131,133,137,139,142,145,149,151,157,163,166,167,169,171

%N Numbers whose divisors are represented by an integer polynomial.

%C 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.

%C 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?

%H T. D. Noe, <a href="/A167857/b167857.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%o (PARI) is(n)=my(d=divisors(n));denominator(content(polinterpolate([0..#d-1],d))) == 1 \\ _Charles R Greathouse IV_, Jan 29 2016

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

%Y Cf. A002476 (primes of the form 6k+1)

%Y Cf. A132230 (primes of the form 30k+1)

%Y Cf. A073103 (primes of the form 210k+1)

%Y Cf. A073917 (least prime of the form k*prime(n)#+1)

%K nonn

%O 1,2

%A _T. D. Noe_, Nov 13 2009