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%I #44 May 10 2021 07:06:58
%S 1,6,15,35,36,77,90,143,210,216,221,225,323,437,462,525,540,667,858,
%T 899,1147,1155,1225,1260,1296,1326,1350,1517,1763,1938,2021,2145,2491,
%U 2622,2695,2772,3127,3150,3240,3315,3375,3599,4002,4087,4757,4845,5005,5148,5183
%N Products of terms of A006094 (products of 2 successive primes).
%C Multiplicative monoid generated by products of two successive primes.
%C 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).
%C The smallest subset A of the positive integers such that
%C 1) 1 is in A
%C 2) if n is in A then so is n * prime(i) * prime(i+1) for all i.
%C Subsequence of A028260.
%C 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
%H Robert Israel, <a href="/A245630/b245630.txt">Table of n, a(n) for n = 1..6742</a>
%H Paul Erdős, <a href="http://www.jstor.org/stable/2306529">Solution to Advanced Problem 4413</a>, American Mathematical Monthly, 59 (1952) 259-261.
%F 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).
%e 1 is in the sequence.
%e 6 = 2*3 is in the sequence.
%e 36 = (2*3)^2 is in the sequence.
%e 90 = (2*3) * (3*5) is in the sequence.
%p N:= 10^6: # to get all terms <= N
%p PP:= [seq(ithprime(i)*ithprime(i+1),i=1.. numtheory[pi](floor(sqrt(N)))-1)]:
%p ext:= (x,p) -> seq(x*p^i,i=0..floor(log[p](N/x))):
%p S:= {1}:
%p for i from 1 to nops(PP) do S:= map(ext,S,PP[i]) od:
%p S;
%t M = 10^6;
%t T = Table[Prime[n] Prime[n + 1], {n, 1, PrimePi[Sqrt[M]]}];
%t T2 = Select[Join[T, T^2], # <= M &];
%t 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 *)
%o (PARI) f(n) = prime(n)*prime(n+1); \\ A006094
%o mul(x,y) = x*y;
%o 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
%Y Cf. A006094, A030229, A059897, A245636.
%Y Subsequence of: A028260, A325698.
%K nonn
%O 1,2
%A _Robert Israel_, Jul 27 2014
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