cp's OEIS Frontend

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.

%I A229094 #38 Mar 31 2021 02:39:40
%S A229094 105,231,627,897,935,1365,1581,1729,2465,2967,4123,4301,4715,5313,
%T A229094 5487,6045,7293,7685,7881,7917,9717,10707,10965,11339,12597,14637,
%U A229094 14993,16377,16445,17353,18753,20213,20757,20915,21045,23779,25327,26331,26765,26961
%N 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.
%C A229094 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.
%H A229094 Amiram Eldar, <a href="/A229094/b229094.txt">Table of n, a(n) for n = 1..10000</a>
%H A229094 Florian Luca and Francesco Pappalardi, <a href="http://doi.org/10.4064/aa129-2-5">Composite positive integers with an average prime factor</a>, Acta Arithmetica, Vol. 129, No. 2 (2007), pp. 197-201.
%F A229094 omega(a(n)) > 2. - _David A. Corneth_, May 01 2017
%e A229094 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).
%e A229094 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
%t A229094 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 *)
%o A229094 (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, ", "))));
%Y A229094 Subsequence of A120944.
%Y A229094 Cf. A185642.
%K A229094 nonn
%O A229094 1,1
%A A229094 _Arkadiusz Wesolowski_, Sep 13 2013