

A002110


Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.
(Formerly M1691 N0668)


756



1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070
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OFFSET

0,2


COMMENTS

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, 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 nth sifting process in an Eratosthenes sieve.  Lekraj Beedassy, 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, Jun 11 2005
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.  David W. Wilson, Oct 23 2006
Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
Records in number of distinct prime divisors.  Artur Jasinski, Apr 06 2008
The digital roots of primorial numbers are multiples of 3.  Parthasarathy Nambi, Aug 19 2009
Denominators of the sum of the ratios of consecutive primes (see A094661).  Vladimir Joseph Stephan Orlovsky, Oct 24 2009
The xth root of the xth primorial has a magnitude on the order of its number of factors (ignoring the first trivial primorial 1, i.e., with 2 counted as the 1st).  Bill McEachen, Feb 08 2010
Where record values occur in A001221.  Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N  N*sum(A005867(i)/A002110(i), i = 0..T(sqrt(N))) This can show for example that at least .16*N numbers are prime less than N for 29^2 > N > 23^2.  Ben Paul Thurston, Aug 23 2010
a(A051838(n)) = A116536(n) * A007504(A051838(n)).  Reinhard Zumkeller, Oct 03 2011
These numbers are divisible by their digital roots, which makes the sequence a subsequence of A064807.  Ivan N. Ianakiev, Oct 08 2013
The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. p(n)# is divisible by 3 for n >= 2.  Christian Schulz, Oct 30 2013


REFERENCES

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 AddisonWesley, Reading, MA, 1962, Vol. 1, p. 50.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209210.
A. W. Lin, S. Zhou, A lineartime algorithm for the orbit problem over cyclic groups, http://homepages.inf.ed.ac.uk/v1awidja/papers/concur14.pdf, 2014
J.L. Nicholas, Petites valeurs de la fonction d'Euler, J. Number Theory 17 (1983), pp. 375388.
P. Ribenboim, The Book of Prime Number Records. SpringerVerlag, 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).
D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 7374.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
C. K. Caldwell, The Prime Glossary, primorial
G. Caveney, J.L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359384.
F. Ellermann, Illustration for A002110, A005867, A038110, A060753
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209210.
A. W. Lin, S. Zhou, A lineartime algorithm for the orbit problem over cyclic groups, 2014
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, 2012.
J.L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375388.
Patrick Sole and Michel Planat, The Robin inequality for 7free integers, INTEGERS, 2011, #A65.
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, Primorial
Index to divisibility sequences


FORMULA

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, Dec 14 2007
a(0) = 1, a(n+1) = prime(n)*a(n).  JuriStepan Gerasimov, Oct 15 2010
a(n) = Product[i = 1..n] A000040(i).  Jonathan Vos Post, Jul 17 2008


EXAMPLE

a(9) = 23# = 2*3*5*7*11*13*17*19*23 = 223092870 divides the difference 5283234035979900 in the arithmetic progression of 26 primes A204189.  Jonathan Sondow, Jan 15 2012


MAPLE

A002110 := n > mul(ithprime(i), i=1..n);


MATHEMATICA

FoldList[Times, 1, Prime[Range[20]]]
s = 0; lst = {}; Do[p = Prime[n]; r = Prime[n + 1]; AppendTo[lst, Denominator[s += r/p]], {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 24 2009 *)
primorial[n_] := Product[Prime[i], {i, n}] (* José María Grau Ribas, Feb 15 2010 *)
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)


PROG

(PARI) a(n)=prod(i=1, n, prime(i)) \\ Washington Bomfim, Sep 23 2008
(PARI) { p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) ) } \\ Harry J. Smith, Nov 13 2009
(Haskell)
a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
 Reinhard Zumkeller, Feb 19 2012, May 03 2011
(MAGMA) [1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012
(SAGE) [sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014


CROSSREFS

Cf. A034387, A005235, A006862, A035345, A035346, A057588, A121572, A094348, A003418, A002182, A002201, A072938, A106037, A033188, A049345 (primorial base representation), A061742 (squares), A036691 (compositorial numbers), A087315.
Sequence in context: A096775 A171989 A233438 * A118491 A088257 A058694
Adjacent sequences: A002107 A002108 A002109 * A002111 A002112 A002113


KEYWORD

nonn,easy,nice,core


AUTHOR

N. J. A. Sloane and J. H. Conway


STATUS

approved



