

A129912


Numbers that are products of distinct primorial numbers (see A002110).


26



1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, 510510, 831600, 900900, 970200, 1021020, 1801800, 2910600, 3063060, 5405400
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OFFSET

1,2


COMMENTS

Conjecture: every odd prime p is either adjacent to a term of A129912 or a prime distance q from some term of A129912, where q < p.  Bill McEachen, Jun 03 2010, edited for clarity in Feb 26 2019
The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences kA151799(k) and A151800(k)k are always either 1 or prime, like is also conjectured to hold for A002182 (cf. also the conjecture given in A117825). However, for A025487, which is a supersequence of both sequences, this is not always true: 512 is a member of A025487, but A151800(512) = 521, with 521  512 = 9, which is a composite number.  Antti Karttunen, Feb 26 2019


REFERENCES

CRC Standard Mathematical Tables, 28th Ed., CRC Press


LINKS



FORMULA

Apart from 1 and 2, numbers of the form 2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s), where p(s) is sth prime, k(i)>0 for i=1..s, k(i)k(i1) = 0 or 1 for i=2..s and {k(1),k(2),..,k(s)}=k(1).  Vladeta Jovovic, Jun 14 2007
Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... .  Amiram Eldar, Jun 03 2023


EXAMPLE

For s = 4 there are 8 (generally 2^(s1)) such numbers: 210 = 2*3*5*7, 420 = 2^2*3*5*7 = (2*3*5*7)*2, 1260 = 2^2*3^2*5*7 = (2*3*5*7)*(2*3), 6300 = 2^2*3^2*5^2*7 = (2*3*5*7)*(2*3*5), 2520 = 2^3*3^2*5*7 = (2*3*5*7)*(2*3)*2, 12600 = 2^3*3^2*5^2*7 = (2*3*5*7)*(2*3*5)*2, 37800 = 2^3*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3), 75600 = 2^4*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3)*2.


MATHEMATICA

Clear[f]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m1], m++]; f[m] (* JeanFrançois Alcover, Mar 03 2014 *) (* or *)
pr[n_] := Product[Prime[n + 1  i]^i, {i, n}]; upto[mx_] := Block[{ric, j = 1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1, Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex  1), ip + 1, ex  1]]]]; Sort@ Reap[ Sow[1]; While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]];


PROG

(PARI) is(n)=my(o=valuation(n, 2), t); if(o<1n<2, return(n==1)); n>>=o; forprime(p=3, , t=valuation(n, p); n/=p^t; if(t>o  t<o1, return(0)); if(t==0, return(n==1)); o=t) \\ Charles R Greathouse IV, Oct 22 2015


CROSSREFS



KEYWORD

easy,nonn


AUTHOR

Bill McEachen, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007


EXTENSIONS

I corrected the Potter link to reflect its relocation.  Bill McEachen, Sep 12 2009
I again corrected the Potter link for its relocation  Bill McEachen, May 30 2013


STATUS

approved



