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A129912 Numbers that are products of distinct primorial numbers (see A002110). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: every odd prime number must either be adjacent to or a prime distance away from a primorial or primorial product (the distance will be a prime smaller than the candidate). - Bill McEachen, Jun 03 2010

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

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 s-th prime, k(i)>0 for i=1..s, k(i)-k(i-1) = 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^(s-1)) 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[m-1], m++]; f[m] (* Jean-Fran├žois 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<1||n<2, return(n==1)); n>>=o; forprime(p=3, , t=valuation(n, p); n/=p^t; if(t>o || t<o-1, return(0)); if(t==0, return(n==1)); o=t) \\ Charles R Greathouse IV, Oct 22 2015

CROSSREFS

Subsequence of A025487. Cf. A002110.

Sequence A283477 sorted into ascending order.

Sequence in context: A166456 A162214 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

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Last modified December 17 08:24 EST 2018. Contains 318192 sequences. (Running on oeis4.)