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A129912 Numbers that are products of distinct primorial numbers (see A002110).

%I A129912 #57 Oct 30 2023 07:39:47
%S A129912 1,2,6,12,30,60,180,210,360,420,1260,2310,2520,4620,6300,12600,13860,
%T A129912 27720,30030,37800,60060,69300,75600,138600,180180,360360,415800,
%U A129912 485100,510510,831600,900900,970200,1021020,1801800,2910600,3063060,5405400
%N A129912 Numbers that are products of distinct primorial numbers (see A002110).
%C A129912 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
%C A129912 The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences k-A151799(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
%D A129912 CRC Standard Mathematical Tables, 28th Ed., CRC Press
%H A129912 Giovanni Resta, <a href="/A129912/b129912.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A129912 Bill McEachen, <a href="http://commons.wikimedia.org/wiki/File:OEIS_A129912_spin1.svg">Normalized A129912</a>.
%H A129912 Robert Potter, <a href="http://primorialconjecture.wordpress.com/2012/07/">Primorial Conjecture</a>.
%H A129912 John Sokol, <a href="http://www.dnull.com/~sokol/prime/conjecture1.html">Sokol's Prime Conjecture</a>, 2002.
%H A129912 Wikipedia, <a href="http://en.wikipedia.org/wiki/Primorial">Primorial</a>.
%H A129912 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>.
%F A129912 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
%F A129912 Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... . - _Amiram Eldar_, Jun 03 2023
%e A129912 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.
%t A129912 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-François Alcover_, Mar 03 2014 *) (* or *)
%t A129912 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]]];
%t A129912 upto[10^30] (* faster, _Giovanni Resta_, Apr 02 2017 *)
%o A129912 (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
%Y A129912 Subsequence of A025487. Sequence A283477 sorted into ascending order.
%Y A129912 Cf. A002110, A117825, A151799, A151800.
%K A129912 easy,nonn
%O A129912 1,2
%A A129912 _Bill McEachen_, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007
%E A129912 Edited by _N. J. A. Sloane_, Jun 09 2007, Aug 08 2007
%E A129912 I corrected the Potter link to reflect its relocation. - _Bill McEachen_, Sep 12 2009
%E A129912 I added link to Wikicommons image. - _Bill McEachen_, Sep 16 2009
%E A129912 I again corrected the Potter link for its relocation - _Bill McEachen_, May 30 2013