

A245630


Products of members of A006094 (products of 2 successive primes).


5



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
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OFFSET

1,2


COMMENTS

Multiplicative monoid generated by products of two successive primes.
All natural numbers 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 natural numbers 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.
Subset 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) 259261.


FORMULA

As n > infinity, a(n)/n^2 > Product_{i>=1} (1  1/sqrt(prime(i)*prime(i+1)) )^2 / (11/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}]] (* JeanFrançois Alcover, Apr 12 2019 *)


CROSSREFS

Cf. A006094, A030229, A059897, A245636.
Subsequence of: A028260, A325698.
Sequence in context: A332735 A120849 A030661 * A049728 A038666 A075625
Adjacent sequences: A245627 A245628 A245629 * A245631 A245632 A245633


KEYWORD

nonn


AUTHOR

Robert Israel, Jul 27 2014


STATUS

approved



