OFFSET
1,2
COMMENTS
The union of products of any number of consecutive odd primes and twice products of any number of consecutive odd primes.
A073485 lists the squarefree numbers with no gaps in their prime factors >= prime(1), and {a(n)} lists the squarefree numbers with no gaps in their prime factors >= prime(2). If we let {b(n)} be the squarefree numbers with no gaps in their prime factors >= prime(3), ..., and let {x(n)} be the squarefree numbers with no gaps in their prime factors >= prime(y), ..., then A073485(n) >= a(n) >= b(n) >= ... >= x(n) >= ... >= A005117(n). [edited by Jon E. Schoenfield, May 26 2018]
Conjecture: if z(n) is the smallest y such that n*k - k^2 is a squarefree number with no gaps in their prime factors >= prime(y) for some k < n, then z(n) >= 1 for all n > 1.
The terms a(n) for which a(n-1) + 1 = a(n) = a(n+1) - 1 begin 2, 6, 14, 30, 106, ... [corrected by Jon E. Schoenfield, May 26 2018]
Squarefree numbers for which any two neighboring odd prime factors in the ordered list of prime factors are consecutive primes. - Felix Fröhlich, Nov 01 2017
EXAMPLE
70 is in this sequence because 2*5*7 = 70 is a squarefree number with two consecutive odd prime factors, 5 and 7.
MAPLE
N:= 1000: # to get all terms <= N
R:= 1, 2:
Oddprimes:= select(isprime, [seq(i, i=3..N, 2)]):
for i from 1 to nops(Oddprimes) do
p:= 1:
for k from i to nops(Oddprimes) do
p:= p*Oddprimes[k];
if p > N then break fi;
if 2*p <= N then R:= R, p, 2*p
else R:= R, p
fi
od;
od:
R:= sort([R]); # Robert Israel, Nov 01 2017
MATHEMATICA
Select[Range@ 166, And[Union@ #2 == {1}, Or[# == {1}, # == {}] &@ Union@ Differences@ PrimePi@ DeleteCases[#1, 2]] & @@ Transpose@ FactorInteger[#] &] (* Michael De Vlieger, Nov 01 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Oct 31 2017
EXTENSIONS
Definition corrected by Michel Marcus, Nov 01 2017
STATUS
approved