cp's OEIS Frontend

A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

%I A002110 M1691 N0668 #387 Aug 15 2025 11:20:17
%S A002110 1,2,6,30,210,2310,30030,510510,9699690,223092870,6469693230,
%T A002110 200560490130,7420738134810,304250263527210,13082761331670030,
%U A002110 614889782588491410,32589158477190044730,1922760350154212639070,117288381359406970983270,7858321551080267055879090
%N A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.
%C A002110 See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.
%C A002110 a(n) is the least number N with n distinct prime factors (i.e., omega(N) = n, cf. A001221). - _Lekraj Beedassy_, Feb 15 2002
%C A002110 Phi(n)/n is a new minimum for each primorial. - _Robert G. Wilson v_, Jan 10 2004
%C A002110 Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - _Lekraj Beedassy_, Mar 31 2005
%C A002110 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
%C A002110 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
%C A002110 Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
%C A002110 Records in number of distinct prime divisors. - _Artur Jasinski_, Apr 06 2008
%C A002110 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]
%C A002110 Denominators of the sum of the ratios of consecutive primes (see A094661). - _Vladimir Joseph Stephan Orlovsky_, Oct 24 2009
%C A002110 Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
%C A002110 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_{i = 0..T(sqrt(N))} A005867(i)/A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - _Ben Paul Thurston_, Aug 23 2010
%C A002110 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
%C A002110 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 "near-primorials" 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
%C A002110 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
%C A002110 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
%C A002110 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
%C A002110 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
%C A002110 The term "primorial" was coined by Harvey Dubner (1987). - _Amiram Eldar_, Apr 16 2021
%C A002110 a(n)^(1/n) is approximately (n log n)/e. - _Charles R Greathouse IV_, Jan 03 2023
%C A002110 Subsequence of A267124. - _Frank M Jackson_, Apr 14 2023
%D A002110 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.
%D A002110 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 49.
%D A002110 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
%D A002110 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002110 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002110 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.
%D A002110 D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.
%H A002110 Alex Ermolaev, <a href="/A002110/b002110.txt">Table of n, a(n) for n = 0..350</a> (terms up to a(100) from T. D. Noe)
%H A002110 Iskander Aliev, Jesús De Loera, Fritz Eisenbrand, Timm Oertel, and Robert Weismantel, <a href="https://arxiv.org/abs/1712.08923">The Support of Integer Optimal Solutions</a>, arXiv:1712.08923 [math.OC], 2017.
%H A002110 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Primorial">Primorial</a>.
%H A002110 Geoffrey Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
%H A002110 Harvey Dubner, <a href="/A006794/a006794.pdf">Factorial and primorial primes</a>, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197-203. (Annotated scanned copy)
%H A002110 F. Ellermann, <a href="/A005867/a005867.txt">Illustration for A002110, A005867, A038110, A060753</a>.
%H A002110 S. W. Golomb, <a href="http://www.jstor.org/stable/2689634">The evidence for Fortune's conjecture</a>, Math. Mag. 54 (1981), 209-210.
%H A002110 D. J. Greenhoe, <a href="https://peerj.com/preprints/520v1.pdf">MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing</a>, 2014.
%H A002110 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858762_Frames_and_Bases_Structure_and_Design_version_020">Frames and Bases: Structure and Design</a>, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, pp. 7, 81.
%H A002110 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858659_A_Book_Concerning_Transforms_version_010">A Book Concerning Transforms</a>, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 7.
%H A002110 A. W. Lin and S. Zhou, <a href="http://homepages.inf.ed.ac.uk/v1awidja/papers/concur14.pdf">A linear-time algorithm for the orbit problem over cyclic groups</a>, preprint, CONCUR 2014 - Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp. 327-341.
%H A002110 A. W. Lin and S. Zhou, <a href="http://dx.doi.org/10.1007/978-3-662-44584-6_23">A linear-time algorithm for the orbit problem over cyclic groups</a>, CONCUR 2014 - Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp. 327-341.
%H A002110 F. E. Masat, <a href="/A005867/a005867_1.pdf">Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991</a>.
%H A002110 R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv:1202.3670 [math.HO], 2012.
%H A002110 Thomas Morrill, <a href="https://arxiv.org/abs/1804.08067">Further Development of "Non-Pythagorean" Musical Scales Based on Logarithms</a>, arXiv:1804.08067 [math.HO], 2018.
%H A002110 J.-L. Nicolas, <a href="http://dx.doi.org/10.1016/0022-314X(83)90055-0">Petites valeurs de la fonction d'Euler</a>, J. Number Theory 17, no.3 (1983), 375-388.
%H A002110 Patrick Sole and Michel Planat, <a href="http://www.emis.de/journals/INTEGERS/papers/l65/l65.Abstract.html">The Robin inequality for 7-free integers</a>, INTEGERS, 2011, #A65.
%H A002110 Andrew V. Sutherland, <a href="http://groups.csail.mit.edu/cis/theses/sutherland-phd.pdf">Order Computations in Generic Groups</a>, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
%H A002110 G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Compter/Factprim.htm">Primorielle</a>.
%H A002110 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%H A002110 Robert G. Wilson v, <a href="/A007014/a007014.pdf">Letter to N. J. A. Sloane, Jan. 1994</a>.
%H A002110 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A002110 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A002110 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H A002110 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A002110 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
%F A002110 a(n) = A054842(A002275(n)).
%F A002110 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
%F A002110 a(0) = 1, a(n+1) = prime(n)*a(n). - _Juri-Stepan Gerasimov_, Oct 15 2010
%F A002110 a(n) = Product_{i=1..n} A000040(i). - _Jonathan Vos Post_, Jul 17 2008
%F A002110 a(A051838(n)) = A116536(n) * A007504(A051838(n)). - _Reinhard Zumkeller_, Oct 03 2011
%F A002110 A000005(a(n)) = 2^n. - _Carlos Eduardo Olivieri_, Jun 16 2015
%F A002110 a(n) = A035345(n) - A005235(n) for n > 0. - _Jonathan Sondow_, Dec 02 2015
%F A002110 For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)). - _Antti Karttunen_, Aug 30 2016
%F A002110 A054841(a(n)) = A002275(n). - _Michael De Vlieger_, Aug 31 2016
%F A002110 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
%F A002110 Sum_{n>=1} 1/a(n) = A064648. - _Amiram Eldar_, Oct 16 2020
%F A002110 Sum_{n>=1} (-1)^(n+1)/a(n) = A132120. - _Amiram Eldar_, Apr 12 2021
%F A002110 Theta being Chebyshev's theta function, a(0) = exp(theta(1)), and for n > 0, a(n) = exp(theta(m)) for A000040(n) <= m < A000040(n+1) where m is an integer. - _Miles Englezou_, Nov 26 2024
%e A002110 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
%p A002110 A002110 := n -> mul(ithprime(i),i=1..n);
%t A002110 FoldList[Times, 1, Prime[Range[20]]]
%t A002110 primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial,20] (* _José María Grau Ribas_, Feb 15 2010 *)
%t A002110 Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* _Harvey P. Dale_, Apr 11 2012 *)
%o A002110 (Haskell)
%o A002110 a002110 n = product $ take n a000040_list
%o A002110 a002110_list = scanl (*) 1 a000040_list
%o A002110 -- _Reinhard Zumkeller_, Feb 19 2012, May 03 2011
%o A002110 (Magma) [1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // _Bruno Berselli_, Oct 24 2012
%o A002110 (Magma) [1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // _Bruno Berselli_, Feb 08 2015
%o A002110 (PARI) a(n)=prod(i=1,n, prime(i)) \\ _Washington Bomfim_, Sep 23 2008
%o A002110 (PARI) p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) )  \\ _Harry J. Smith_, Nov 13 2009
%o A002110 (PARI) a(n) = factorback(primes(n)) \\ _David A. Corneth_, May 06 2018
%o A002110 (Python)
%o A002110 from sympy import primorial
%o A002110 def a(n): return 1 if n < 1 else primorial(n)
%o A002110 [a(n) for n in range(51)]  # _Indranil Ghosh_, Mar 29 2017
%o A002110 (Sage) [sloane.A002110(n) for n in (1..20)] # _Giuseppe Coppoletta_, Dec 05 2014
%o A002110 (Scheme) ; with memoization-macro definec
%o A002110 (definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; _Antti Karttunen_, Aug 30 2016
%Y A002110 A034386 gives the second version of the primorial numbers.
%Y A002110 Subsequence of A005117 and of A064807. Apart from the first term, a subsequence of A083207.
%Y A002110 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.
%Y A002110 Cf. A061720 (first differences), A143293 (partial sums).
%Y A002110 Cf. also A276085, A276086.
%Y A002110 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.
%K A002110 nonn,easy,nice,core
%O A002110 0,2
%A A002110 _N. J. A. Sloane_ and _J. H. Conway_