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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 (list; graph; refs; listen; history; text; internal format)
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) 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 *)

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

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Last modified August 12 09:24 EDT 2020. Contains 336438 sequences. (Running on oeis4.)