0,2

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

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.

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

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

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

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

C. K. Caldwell, The Prime Glossary, primorial

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

G. Villemin's Almanach of Numbers, Primorielle

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

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

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);

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*) - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008

(PARI) a(n)=prod(i=1, n, prime(n))

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

Primorial base representation: A049345.

Cf. A136104, A121572.

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

njas and J. H. Conway (conway(AT)math.princeton.edu)