

A002110


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


1478



1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090
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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.
a(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
For n>=2, the digital roots of a(n) are multiples of 3.  Parthasarathy Nambi, Aug 19 2009 (With corrections by Zak Seidov, Aug 30 2015).
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. prime(n)# is divisible by 3 for n >= 2.  Christian Schulz, Oct 30 2013
The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "nearprimorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3.  Richard R. Forberg, Jul 01 2015
Subsequence of A005117.  Michel Marcus, Feb 22 2016
For n>1, a(n) is a Zumkeller number (A083207).  Ivan N. Ianakiev, May 08 2016
If a number k and a(n) are coprime and k < (prime(n+1))^b < a(n), where b is an integer, then k has fewer than b prime factors, counting multiplicity (i.e., bigomega(k) < b, cf. A001222).  Isaac Saffold, Dec 03 2017
If n > 0, then a(n) has 2^n unitary divisors (A034444), and a(n) is a record; i.e., if k < a(n) then k has fewer unitary divisors than a(n) has.  Clark Kimberling, Jun 26 2018
Unitary superabundant numbers: numbers k with a record value of the unitary abundancy index, A034448(k)/k > A034448(m)/m for all m < k.  Amiram Eldar, Apr 20 2019
Psi(n)/n is a new maximum for each primorial (psi = A001615) [proof in link: Patrick Sole and Michel Planat, proposition 1 page 2]; compare with comment 2004: Phi(n)/n is a new minimum for each primorial.  Bernard Schott, May 21 2020
The term "primorial" was coined by Harvey Dubner (1987).  Amiram Eldar, Apr 16 2021


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.
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

Alex Ermolaev, Table of n, a(n) for n = 0..350 (terms up to a(100) from T. D. Noe)
Iskander Aliev, Jesús De Loera, Fritz Eisenbrand, Timm Oertel, and Robert Weismantel, The Support of Integer Optimal Solutions, arXiv:1712.08923 [math.OC], 2017.
C. K. Caldwell, The Prime Glossary, primorial
G. Caveney, J.L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 20112012; Ramanujan J., 29 (2012), 359384.
Harvey Dubner, Factorial and primorial primes, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197203. (Annotated scanned copy)
F. Ellermann, Illustration for A002110, A005867, A038110, A060753
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209210.
D. J. Greenhoe, MRAWavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, pp. 7, 81.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 7.
A. W. Lin and S. Zhou, A lineartime algorithm for the orbit problem over cyclic groups, preprint, CONCUR 2014  Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp 327341.
A. W. Lin and S. Zhou, A lineartime algorithm for the orbit problem over cyclic groups, CONCUR 2014  Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp 327341.
F. E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv:1202.3670 [math.HO], 2012.
Thomas Morrill, Further Development of "NonPythagorean" Musical Scales Based on Logarithms, arXiv:1804.08067 [math.HO], 2018.
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
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers


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
A000005(a(n)) = 2^n.  Carlos Eduardo Olivieri, Jun 16 2015
a(n) = A035345(n)  A005235(n) for n > 0.  Jonathan Sondow, Dec 02 2015
For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)).  Antti Karttunen, Aug 30 2016
A054841(a(n)) = A002275(n)  Michael De Vlieger, Aug 31 2016
a(n) = A270592(2*n+2)  A270592(2*n+1) if 0 <= n <= 4 (conjectured for all n by Alon Kellner).  Jonathan Sondow, Mar 25 2018
Sum_{n>=1} 1/a(n) = A064648.  Amiram Eldar, Oct 16 2020
Sum_{n>=1} (1)^(n+1)/a(n) = A132120.  Amiram Eldar, Apr 12 2021


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]]]
primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial, 20] (* José María Grau Ribas, Feb 15 2010 *)
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)


PROG

(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
(Magma) [1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015
(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
(PARI) a(n) = factorback(primes(n)) \\ David A. Corneth, May 06 2018
(Python)
from sympy import primorial
def a(n): return 1 if n < 1 else primorial(n)
[a(n) for n in range(51)] # Indranil Ghosh, Mar 29 2017
(Sage) [sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014
(Scheme) ; with memoizationmacro definec
(definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 ( n 1))))) ;; Antti Karttunen, Aug 30 2016


CROSSREFS

Cf. A001615, A002182, A002201, A003418, A005235, A006862, A034444 (unitary divisors), A034448, A034387, A033188, A035345, A035346, A036691 (compositorial numbers), A049345 (primorial base representation), A057588, A060735 (and integer multiples), A061742 (squares), A072938, A079266, A087315, A094348, A106037, A121572, A053589, A064648, A132120, A260188.
Cf. A061720 (first differences), A143293 (partial sums).
Cf. also A276085, A276086.
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.
Sequence in context: A171989 A335069 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



