

A287935


a(n) is the smallest squarefree number k with n prime factors such that gcd(k, d2d1) = 1 for all coprime pairs of divisors of k, 1 < d1 < d2 < k.


0




OFFSET

1,1


COMMENTS

a(2)  a(5) were calculated by Erdős and Evans. They formulated the sequence in terms of sets of primes. They proved that a(n) exists for all n >= 2, and that 12/n < log2(a(n))/n^2 < log2(3 + e), where the upper bound holds for any e > 0 for sufficiently large n.
The prime factors of the first terms are:
a(1) = 2
a(2) = 2 * 3
a(3) = 3 * 5 * 7
a(4) = 7 * 11 * 13 * 23
a(5) = 13 * 17 * 29 * 41 * 61
a(6) = 19 * 37 * 47 * 53 * 101 * 229
a(7) <= 1378987557700217.  Giovanni Resta, Jun 03 2017


LINKS

Table of n, a(n) for n=1..6.
Paul Erdős and Anthony B. Evans, Sets of Prime Numbers Satisfying a Divisibility Condition, Journal of Number Theory, Vol. 61, No. 1, (1996), pp. 3943.


EXAMPLE

105 = 3 * 5 * 7, gcd(53, 105) = gcd(73, 105) = gcd(75, 105) = gcd(3*57, 105) = gcd(3*75, 105) = gcd(5*73, 105) = 1. 105 is the smallest product of 3 different primes with this property, thus a(3) = 105.


MATHEMATICA

aQ[n_] := Module[{g = True}, d = Drop[Drop[Divisors[n], 1], 1]; nd = Length[d]; For[k1 = 0, k1 < nd  1, k1++; g1 = 1; d1 = d[[k1]]; For[k2 = k1, k2 < nd, k2++; d2 = d[[k2]]; If[GCD[d1, d2] > 1, Continue[]]; If[GCD[n, d2  d1] > 1, g1 = 0; Break[]]]; If[g1 == 0, g = False; Break[]]]; g]; m = 2; k = 1; a = {}; While[Length[a] < 5, If[SquareFreeQ[k] && PrimeNu[k] == m && aQ[k], a = AppendTo[a, k]; m++, k++]]; a


CROSSREFS

Sequence in context: A099790 A294906 A284262 * A181036 A222854 A059088
Adjacent sequences: A287932 A287933 A287934 * A287936 A287937 A287938


KEYWORD

nonn,more


AUTHOR

Amiram Eldar, Jun 03 2017


STATUS

approved



