A245630 Products of terms of A006094 (products of 2 successive primes).

1, 6, 15, 35, 36, 77, 90, 143, 210, 216, 221, 225, 323, 437, 462, 525, 540, 667, 858, 899, 1147, 1155, 1225, 1260, 1296, 1326, 1350, 1517, 1763, 1938, 2021, 2145, 2491, 2622, 2695, 2772, 3127, 3150, 3240, 3315, 3375, 3599, 4002, 4087, 4757, 4845, 5005, 5148, 5183

Offset: 1

Author

Robert Israel, Jul 27 2014

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A006094, A030229, A059897, A245636, A267251.

Subsequence of: A028260, A325698.

Programs

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 *)

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->(xMichel Marcus, May 10 2021

Formula

Limit_{n->oo} a(n)/n^2 = Product_{i>=1} (1 - 1/sqrt(prime(i)*prime(i+1)))^2 / (1 - 1/prime(i))^2 = 1/A267251^2 (see Erdős reference).