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A002110
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Primorial numbers (first definition): product of first n primes. Sometimes written p(n)#.
(Formerly M1691 N0668)
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644
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1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (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 (JudMcCranie(AT)ugaalum.uga.edu), 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
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]
The xth root of the xth primorial has a magnitude on the order of its number of factors (ignoring the first trivial primorial 1 ie with 2 counted as the 1st). [From Bill R McEachen (bmceache(AT)centralsan.org), Feb 08 2010]
Where record values occur in A001221 [From 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 [From Ben Thurston (benpaulthurston(AT)gmail.com), Aug 23 2010]
Partial products of non-composite numbers. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 15 2010]
a(A051838(n)) = A116536(n) * A007504(A051838(n)). [Reinhard Zumkeller, Oct 03 2011]
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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 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).
Patrick Sole and Michel Planat, THE ROBIN INEQUALITY FOR 7-FREE INTEGERS, INTEGERS, 2011, #A65; http://www.emis.de/journals/INTEGERS/papers/l65/l65.pdf
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.
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LINKS
| 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
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
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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 (qntmpkt(AT)yahoo.com), Dec 14 2007
a(1)=1, a(n+1)=prime(n)*a(n). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 15 2010]
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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
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MAPLE
| 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]
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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 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 24 2009]
primorial[n_] := Product[Prime[i], {i, n}] [From J.M. Grau Ribas (grau(AT)uniovi.es), Feb 15 2010]
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PROG
| (PARI) a(n)=prod(i=1, n, prime(i)) - W. Bomfim (webonfim(AT)bol.com.br), Sep 23 2008
(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]
(Haskell)
a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
-- Reinhard Zumkeller, Feb 19 2012, May 03 2011
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CROSSREFS
| 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.
Cf. A036691 (compositorial numbers).
Cf. A033188, A204189.
Sequence in context: A068215 A096775 A171989 * A118491 A088257 A058694
Adjacent sequences: A002107 A002108 A002109 * A002111 A002112 A002113
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KEYWORD
| nonn,easy,nice,core,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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