

A229094


Composite squarefree numbers k such that the arithmetic mean of the distinct prime factors of k is a prime p, and p divides k.


2



105, 231, 627, 897, 935, 1365, 1581, 1729, 2465, 2967, 4123, 4301, 4715, 5313, 5487, 6045, 7293, 7685, 7881, 7917, 9717, 10707, 10965, 11339, 12597, 14637, 14993, 16377, 16445, 17353, 18753, 20213, 20757, 20915, 21045, 23779, 25327, 26331, 26765, 26961
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
omega(a(n)) > 2.  David A. Corneth, May 01 2017


LINKS

Table of n, a(n) for n=1..40.
Florian Luca and Francesco Pappalardi, Composite positive integers with an average prime factor, Acta Arithmetica 129 (2007), pp. 197201.


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]] (* JeanFrançois 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

Cf. A185642. Subsequence of A120944.
Sequence in context: A176878 A088595 A308643 * A307108 A262723 A250757
Adjacent sequences: A229091 A229092 A229093 * A229095 A229096 A229097


KEYWORD

nonn


AUTHOR

Arkadiusz Wesolowski, Sep 13 2013


STATUS

approved



