A231814 - OEIS

%I #18 Mar 24 2023 17:01:53

%S 6,15,30,91,703,1891,2701,12403,18721,38503,49141,51319,79003,88831,

%T 104653,146611,188191,218791,226801,269011,286903,385003,497503,

%U 597871,665281,721801,736291,765703,873181,954271,1056331,1314631,1373653,1537381,1755001,1869211

%N Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

%C Squarefree numbers with k >= 2 prime factors of the form p_1 * p_2 * ... * p_k, where p_1 < p_2 < ... < p_k = primes with p_k = 2 * p_(k-1) - 1.

%C Each of these numbers is divisible by the arithmetic mean of its proper divisors.

%C Supersequence of A129521 (numbers of the form p*q, p and q prime with q=2*p-1; see A005382 and A005383).

%H Robert Israel, <a href="/A231814/b231814.txt">Table of n, a(n) for n = 1..10000</a>

%e 51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.

%p N:= 10^7: # for terms <= N

%p p:= 1: S:= NULL: count:= 0:

%p do

%p   p:= nextprime(p);

%p   if p*(2*p-1) > N then break fi;

%p   q:= p; x:= p;

%p   do

%p     q:= 2*q-1;

%p     if not isprime(q) then break fi;

%p     x:= x*q;

%p     if x > N then break fi;

%p     S:= S,x; count:= count+1;

%p   od;

%p od:

%p sort([S]); # _Robert Israel_, Mar 24 2023

%t geomQ[lst_] := Module[{x = lst - 1}, x = x/x[[1]]; Log[2, x] + 1 == Range[Length[x]]]; Select[Range[2, 1000000], ! PrimeQ[#] && SquareFreeQ[#] && geomQ[Transpose[FactorInteger[#]][[1]]] &] (* _T. D. Noe_, Nov 14 2013 *)

%Y Cf. A000040, A005117, A005382, A005383, A129521, A231815, A231816.

%Y Cf. A057330 (first prime for such numbers that has n factors).

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Nov 13 2013