

A129912


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


12



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

T. D. Noe and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Bill McEachen, Normalized A129912
Robert Potter, Primorial Conjecture.
J. Sokol, Sokol's Prime Conjecture
Wikipedia, Primorial
Index entries for sequences related to primorial numbers


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


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]]];
upto[10^30] (* faster, Giovanni Resta, Apr 02 2017 *)


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

Subsequence of A025487. Sequence A283477 sorted into ascending order.
Cf. A002110, A117825, A151799, A151800.
Sequence in context: A162214 A309728 A100071 * A283477 A182863 A161507
Adjacent sequences: A129909 A129910 A129911 * A129913 A129914 A129915


KEYWORD

easy,nonn


AUTHOR

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


EXTENSIONS

Edited by N. J. A. Sloane, Jun 09 2007, Aug 08 2007
I corrected the Potter link to reflect its relocation.  Bill McEachen, Sep 12 2009
I added link to Wikicommons image.  Bill McEachen, Sep 16 2009
I again corrected the Potter link for its relocation  Bill McEachen, May 30 2013


STATUS

approved



