|
|
A231814
|
|
Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.
|
|
3
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.
|
|
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:
|
|
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 *)
|
|
CROSSREFS
|
Cf. A057330 (first prime for such numbers that has n factors).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|