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

1,2

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

Cf. A006094, A030229, A059897, A245636.

Subsequence of: A028260, A325698.

Sequence in context: A358210 A338053 A030661 * A357848 A049728 A038666

Adjacent sequences: A245627 A245628 A245629 * A245631 A245632 A245633

Keyword

nonn

Author

Robert Israel, Jul 27 2014

Status

approved