%I A002110 M1691 N0668
%S A002110 1,2,6,30,210,2310,30030,510510,9699690,223092870,6469693230,
%T A002110 200560490130,7420738134810,304250263527210,13082761331670030,
%U A002110 614889782588491410,32589158477190044730,1922760350154212639070
%N A002110 Primorial numbers (first definition): product of first n primes. Sometimes 
               written p#.
%C A002110 See A034386 for the second definition of primorial numbers: product of 
               primes in the range 2 to n.
%C A002110 p(n)# is the least number N with n distinct prime factors (i.e. omega(N)=n, 
               cf. A001221). - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 15 2002
%C A002110 Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v Jan 
               10 2004.
%C A002110 Smallest number stroked off n times after the n-th sifting process in 
               an Eratosthenes sieve. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Mar 31 2005
%C A002110 Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/pi^2 
               < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie (j.mccranie(AT)comcast.net), 
               Jun 11 2005
%C A002110 Comment from David W. Wilson (davidwwilson(AT)comcast.net), Oct 23 2006: 
               Let f be a multiplicative function with f(p) > f(p^k) > 1 (p prime, 
               k > 1), f(p) > f(q) > 1 (p, q prime, p < q). Then the record maxima 
               of f occur at n# for n >= 1. Similarly, if 0 < f(p) < f(p^k) < 1 
               (p prime, k > 1), 0 < f(p) < f(q) < 1 (p, q prime, p < q), then the 
               record minima of f occur at n# for n >= 1.
%C A002110 Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
%C A002110 Records in number of distinct prime divisors - Artur Jasinski (grafix(AT)csl.pl), 
               Apr 06 2008
%C A002110 Carella proves on p. 12 what J.-L. Nicholas asserted in 1983, namely 
               that, if the Riemann Hypothesis is true, a(n)/phi(a(n)) > (e^gamma) 
               log log a(n) for all sufficiently large a(n), where phi is the Euler 
               totient function A000010. Conversely, if the Riemann Hypothesis is 
               false, then both a(n)/phi(a(n)) > (e^gamma) log log a(n) and a(n)/
               phi(a(n)) < (e^gamma) log log a(n) occur for infinitely many k => 
               1. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008. Warning: 
               See the following comments! - N. J. A. Sloane, Jul 21 2009
%C A002110 Comments from Geoffrey Caveney (rokirovka(AT)gmail.com), May 17 2009: 
               (Start)
%C A002110 (Responding to the comment of Jonathan Vos Post about the paper of Carella 
               referenced in the notes.)
%C A002110 The theorem Carella *claims* to prove (his Theorem 7), if true, would
%C A002110 actually amount to a proof of the Riemann Hypothesis when combined with 
               the
%C A002110 theorem of Nicolas (Theorem 6 in Carella's paper):
%C A002110 On page 2 Carella states as Theorem 6 Nicolas' result that (i) if the
%C A002110 Riemann Hypothesis is true, then N_k / phi(N_k) > e^gamma log log(N_k) 
               for
%C A002110 all k >= 1, and (ii) if the Riemann Hypothesis is false, then both N_k 
               /
%C A002110 phi(N_k) < e^gamma log log(N_k) and N_k / phi(N_k) > e^gamma log log(N_k)
%C A002110 occur for infinitely many k >= 1.
%C A002110 Then Carella states as Theorem 7 his own result that N_k / phi(N_k) >
%C A002110 e^gamma log log(N_k) for all sufficiently large integer N_k. He presents 
               his
%C A002110 claimed proof of this result on pages 12-13.
%C A002110 But Carella's paper does not seem to note the fact that if his Theorem 
               7 is
%C A002110 true and Nicolas' Theorem 6 is true, this amounts to a proof of the Riemann 
               Hypothesis:
%C A002110 If N_k / phi(N_k) > e^gamma log log(N_k) for all sufficiently large integer
%C A002110 N_k, then there can only be finitely many k such that N_k / phi(N_k) 
               <= e^gamma log log(N_k).
%C A002110 Therefore N_k / phi(N_k) < e^gamma log log(N_k) cannot occur for infinitely 
               many k >= 1.
%C A002110 Therefore by Theorem 6-ii, the Riemann Hypothesis cannot be false. Thus 
               the
%C A002110 Riemann Hypothesis is proved to be true.
%C A002110 One would expect to find a flaw in a one-page proof of a result that 
               implies
%C A002110 the Riemann Hypothesis. Here is the first one:
%C A002110 On page 12 Carella begins his proof as follows:
%C A002110 "On the contrary suppose that N_k / phi(N_k) <= e^gamma log log(N_k). 
               Then
%C A002110 log Product_[p|N_k] (1 - 1/p^2)^-1 (1 + 1/p) <= log(e^gamma) log
%C A002110 log(N_k), (8)
%C A002110 see Proposition 8-i."
%C A002110 There is not, however, any Proposition 8-i to be found in his paper. 
               (End)
%C A002110 Successive minimal records in value of EulerPhi[k]/k. [From Artur Jasinski 
               (grafix(AT)csl.pl), Nov 05 2008]
%C A002110 The digital roots of primorial numbers are multiples of 3. [From Parthasarathy 
               Nambi (PachaNambi(AT)yahoo.com), Aug 19 2009]
%C A002110 Denominators of the sum of the ratios of consecutive primes. Cf. A094661 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 24 2009]
%D A002110 A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index 
               of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford 
               and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
%D A002110 S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 
               209-210.
%D A002110 J.-L. Nicholas, Petites valeurs de la fonction d'Euler, J. Number Theory 
               17(1983)375-388.
%D A002110 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 4.
%D A002110 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002110 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002110 Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, 
               Math. Dept., M.I.T., 2007.
%D A002110 D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, 
               Peters, 2005, pp. 73-74.
%H A002110 T. D. Noe, <a href="b002110.txt">Table of n, a(n) for n = 0..100</a>
%H A002110 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=Primorial">primorial</a>
%H A002110 N. A. Carella, <a href="http://arxiv.org/pdf/0806.3620">Divisor and Totient 
               Functions Estimates</a>
%H A002110 F. Ellermann, <a href="a005867.txt">Illustration for A002110, A005867, 
               A038110, A060753</a>
%H A002110 Andrew V. Sutherland, <a href="http://groups.csail.mit.edu/cis/theses/
               sutherland-phd.pdf">Order Computations in Generic Groups</a>, Ph. 
               D. Dissertation, Math. Dept., M.I.T., 2007.
%H A002110 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
               Compter/Factprim.htm">Primorielle</a>
%H A002110 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Primorial.html">Link to a section of The World of Mathematics.</a>
%F A002110 Asymptotic expression for a(n): exp((1 + o(1)) * n * log(n)) where o(1) 
               is the "little o" notation - Dan Fux (dan.fux(AT)OpenGaia.com or 
               danfux(AT)OpenGaia.com), Apr 08 2001
%F A002110 a(n) = A054842(A002275(n))
%F A002110 Binomial transform = A136104: (1, 3, 11, 55, 375, 3731,...). Equals binomial 
               transform of A121572: (1, 1, 3, 17, 119, 1509,...). - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 14 2007
%p A002110 A002110 := n->product('ithprime(i )','i'=1..n);
%p A002110 with (numtheory):a:=n->mul(ithprime(j), j=1..n):seq(a(n), n=0..17); [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A002110 FoldList[Times, 1, Prime[Range[20]]]
%t A002110 max = 0; a = {1}; Do[w = Length[FactorInteger[n]]; If[w > max, AppendTo[a, 
               n]; max = w], {n, 2, 100000}]; a - Artur Jasinski (grafix(AT)csl.pl), 
               Apr 06 2008
%t A002110 aa = {}; min = 2; Do[k = EulerPhi[n]/n; If[k < min, AppendTo[aa, n]; 
               min = k], {n, 1, 200000}]; aa [From Artur Jasinski (grafix(AT)csl.pl), 
               Nov 05 2008]
%t A002110 s=0;lst={};Do[p=Prime[n];r=Prime[n+1];AppendTo[lst,Denominator[s+=r/p]],
               {n,3*4!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 
               24 2009]
%o A002110 (PARI) a(n)=prod(i=1,n, prime(i)) - W. Bomfim (webonfim(AT)bol.com.br), 
               Sep 23 2008
%o A002110 (PARI) { p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", 
               n, " ", p) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Nov 13 2009]
%Y A002110 Cf. A034387, A005235, A006862, A035345, A035346, A057588, A136104, A121572.
%Y A002110 Primorial base representation: A049345.
%Y A002110 Squares: A061742.
%Y A002110 a(n) = Product[i=1..n] A000040(i). - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Jul 17 2008
%Y A002110 Cf. A094348, A003418, A002182, A002201, A072938, A106037.
%Y A002110 Sequence in context: A129779 A068215 A096775 this_sequence A118491 A088257 
               A058694
%Y A002110 Adjacent sequences: A002107 A002108 A002109 this_sequence A002111 A002112 
               A002113
%K A002110 nonn,easy,nice,core,new
%O A002110 0,2
%A A002110 N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)