OFFSET
1,2
COMMENTS
Multiplicative monoid generated by products of two successive primes.
All positive integers of the form Product_{i>=1} (prime(i)*prime(i+1))^m_i for integers m_i >= 0 (all but finitely many m_i = 0).
The smallest subset A of the positive integers such that
1) 1 is in A
2) if n is in A then so is n * prime(i) * prime(i+1) for all i.
Subsequence of A028260.
If A059897(.,.) is used as multiplicative operator in place of standard integer multiplication, A006094 generates A030229 (products of an even number of distinct primes). - Peter Munn, Oct 04 2019
LINKS
Robert Israel, Table of n, a(n) for n = 1..6742
Paul Erdős, Solution to Advanced Problem 4413, American Mathematical Monthly, 59 (1952) 259-261.
FORMULA
As n -> infinity, a(n)/n^2 -> Product_{i>=1} (1 - 1/sqrt(prime(i)*prime(i+1)) )^2 / (1-1/prime(i))^2 ) (see Erdős reference).
EXAMPLE
1 is in the sequence.
6 = 2*3 is in the sequence.
36 = (2*3)^2 is in the sequence.
90 = (2*3) * (3*5) is in the sequence.
MAPLE
N:= 10^6: # to get all terms <= N
PP:= [seq(ithprime(i)*ithprime(i+1), i=1.. numtheory[pi](floor(sqrt(N)))-1)]:
ext:= (x, p) -> seq(x*p^i, i=0..floor(log[p](N/x))):
S:= {1}:
for i from 1 to nops(PP) do S:= map(ext, S, PP[i]) od:
S;
MATHEMATICA
M = 10^6;
T = Table[Prime[n] Prime[n + 1], {n, 1, PrimePi[Sqrt[M]]}];
T2 = Select[Join[T, T^2], # <= M &];
Join[{1}, T2 //. {a___, b_, c___, d_, e___} /; b*d <= M && FreeQ[{a, b, c, d, e}, b*d] :> Sort[{a, b, c, d, e, b*d}]] (* Jean-François Alcover, Apr 12 2019 *)
PROG
(PARI) f(n) = prime(n)*prime(n+1); \\ A006094
mul(x, y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok, nv = select(x->(x<lim), Set(concat(v, setbinop(mul, v)))); ok = (#v == #nv); v = nv; ); concat(1, v); } \\ Michel Marcus, May 10 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert Israel, Jul 27 2014
STATUS
approved