OFFSET
1,1
COMMENTS
Let A(x) be the set of terms <= x. The estimates x/(exp((2 + o(1))*sqrt(log x log log x)) <= #A(x) <= x/(exp((1/sqrt(2) + o(1))*sqrt(log x log log x)) hold as x -> infinity.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and Francesco Pappalardi, Composite positive integers with an average prime factor, Acta Arithmetica, Vol. 129, No. 2 (2007), pp. 197-201.
FORMULA
omega(a(n)) > 2. - David A. Corneth, May 01 2017
EXAMPLE
935 is in the list for the following reasons. First, 935 is squarefree and composite. Secondly the distinct prime factors of 935 are 5, 11, and 17, and the average of these three prime factors is 11, which is also prime. Finally, 935 is divisible by 11 (the prime average of the distinct prime factors).
Similarly, 1365 is in the list since it is composite, squarefree, and its distinct prime factors are 3, 5, 7, and 13. The average of the prime factors is 28/4=7, 7 is prime, and 7 divides 1365. - Tom Edgar, Oct 21 2014
MATHEMATICA
Reap[For[k = 6, k < 10^5, k++, If[SquareFreeQ[k] && CompositeQ[k], m = Mean[FactorInteger[k][[All, 1]]]; If[IntegerQ[m] && PrimeQ[m] && Mod[k, m] == 0, Print[k]; Sow[k]]]]][[2, 1]] (* Jean-François Alcover, May 01 2017 *)
PROG
(PARI) for(n=2, 26961, if(issquarefree(n)&&!isprime(n), o=omega(n); s=sum(i=1, o, factor(n)[, 1][i]); a=s/o; if(!frac(a)&&isprime(a)&&!Mod(n, a), print1(n, ", "))));
CROSSREFS
KEYWORD
nonn
AUTHOR
Arkadiusz Wesolowski, Sep 13 2013
STATUS
approved