A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

6, 15, 30, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 51319, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211

Offset: 1

Author

Jaroslav Krizek, Nov 13 2013

Keywords

• nonn

Comments

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.

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

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Programs

Maple

N:= 10^7: # for terms <= N
p:= 1: S:= NULL: count:= 0:
do
  p:= nextprime(p);
  if p*(2*p-1) > N then break fi;
  q:= p; x:= p;
  do
    q:= 2*q-1;
    if not isprime(q) then break fi;
    x:= x*q;
    if x > N then break fi;
    S:= S,x; count:= count+1;
  od;
od:
sort([S]); # Robert Israel, Mar 24 2023

Mathematica

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 *)