0,2

See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.

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

Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v Jan 10 2004.

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

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

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.

Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.

Records in number of distinct prime divisors - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008

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

Comments from Geoffrey Caveney (rokirovka(AT)gmail.com), May 17 2009: (Start)

(Responding to the comment of Jonathan Vos Post about the paper of Carella referenced in the notes.)

The theorem Carella *claims* to prove (his Theorem 7), if true, would

actually amount to a proof of the Riemann Hypothesis when combined with the

theorem of Nicolas (Theorem 6 in Carella's paper):

On page 2 Carella states as Theorem 6 Nicolas' result that (i) if the

Riemann Hypothesis is true, then N_k / phi(N_k) > e^gamma log log(N_k) for

all k >= 1, and (ii) if the Riemann Hypothesis is false, then both N_k /

phi(N_k) < e^gamma log log(N_k) and N_k / phi(N_k) > e^gamma log log(N_k)

occur for infinitely many k >= 1.

Then Carella states as Theorem 7 his own result that N_k / phi(N_k) >

e^gamma log log(N_k) for all sufficiently large integer N_k. He presents his

claimed proof of this result on pages 12-13.

But Carella's paper does not seem to note the fact that if his Theorem 7 is

true and Nicolas' Theorem 6 is true, this amounts to a proof of the Riemann Hypothesis:

If N_k / phi(N_k) > e^gamma log log(N_k) for all sufficiently large integer

N_k, then there can only be finitely many k such that N_k / phi(N_k) <= e^gamma log log(N_k).

Therefore N_k / phi(N_k) < e^gamma log log(N_k) cannot occur for infinitely many k >= 1.

Therefore by Theorem 6-ii, the Riemann Hypothesis cannot be false. Thus the

Riemann Hypothesis is proved to be true.

One would expect to find a flaw in a one-page proof of a result that implies

the Riemann Hypothesis. Here is the first one:

On page 12 Carella begins his proof as follows:

"On the contrary suppose that N_k / phi(N_k) <= e^gamma log log(N_k). Then

log Product_[p|N_k] (1 - 1/p^2)^-1 (1 + 1/p) <= log(e^gamma) log

log(N_k), (8)

see Proposition 8-i."

There is not, however, any Proposition 8-i to be found in his paper. (End)

Successive minimal records in value of EulerPhi[k]/k. [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]

The digital roots of primorial numbers are multiples of 3. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 19 2009]

Denominators of the sum of the ratios of consecutive primes. Cf. A094661 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 24 2009]

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.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

J.-L. Nicholas, Petites valeurs de la fonction d'Euler, J. Number Theory 17(1983)375-388.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.

T. D. Noe, Table of n, a(n) for n = 0..100

C. K. Caldwell, The Prime Glossary, primorial

N. A. Carella, Divisor and Totient Functions Estimates

F. Ellermann, Illustration for A002110, A005867, A038110, A060753

Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

G. Villemin's Almanach of Numbers, Primorielle

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

a(n) = A054842(A002275(n))

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

A002110 := n->product('ithprime(i )', 'i'=1..n);

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]

FoldList[Times, 1, Prime[Range[20]]]

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

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]

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]

(PARI) a(n)=prod(i=1, n, prime(i)) - W. Bomfim (webonfim(AT)bol.com.br), Sep 23 2008

Cf. A034387, A005235, A006862, A035345, A035346, A057588, A136104, A121572.

Primorial base representation: A049345.

Squares: A061742.

a(n) = Product[i=1..n] A000040(i). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008

Cf. A094348, A003418, A002182, A002201, A072938, A106037. [From Ivor C. Quence (Ivan(AT)email_address.too), May 07 2009]

Adjacent sequences: A002107 A002108 A002109 this_sequence A002111 A002112 A002113

Sequence in context: A129779 A068215 A096775 this_sequence A118491 A088257 A058694

nonn,easy,nice,core,new

N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)