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

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

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.

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]

(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

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