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%I M1691 N0668 #373 Jun 04 2024 08:19:13
%S 1,2,6,30,210,2310,30030,510510,9699690,223092870,6469693230,
%T 200560490130,7420738134810,304250263527210,13082761331670030,
%U 614889782588491410,32589158477190044730,1922760350154212639070,117288381359406970983270,7858321551080267055879090
%N Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.
%C See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.
%C 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 Phi(n)/n is a new minimum for each primorial. - _Robert G. Wilson v_, Jan 10 2004
%C Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - _Lekraj Beedassy_, Mar 31 2005
%C 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 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 Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
%C Records in number of distinct prime divisors. - _Artur Jasinski_, Apr 06 2008
%C 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 Denominators of the sum of the ratios of consecutive primes (see A094661). - _Vladimir Joseph Stephan Orlovsky_, Oct 24 2009
%C Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
%C 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 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 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 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 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 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 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 The term "primorial" was coined by Harvey Dubner (1987). - _Amiram Eldar_, Apr 16 2021
%C a(n)^(1/n) is approximately (n log n)/e. - _Charles R Greathouse IV_, Jan 03 2023
%C Subsequence of A267124. - _Frank M Jackson_, Apr 14 2023
%D 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 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.
%H 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 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 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Primorial">Primorial</a>.
%H G. 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 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 F. Ellermann, <a href="/A005867/a005867.txt">Illustration for A002110, A005867, A038110, A060753</a>.
%H 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 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 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 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 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 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 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 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 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 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 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 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 G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Compter/Factprim.htm">Primorielle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%H Robert G. Wilson v, <a href="/A007014/a007014.pdf">Letter to N. J. A. Sloane, Jan. 1994</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F 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 a(n) = A054842(A002275(n)).
%F 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 a(0) = 1, a(n+1) = prime(n)*a(n). - _Juri-Stepan Gerasimov_, Oct 15 2010
%F a(n) = Product_{i=1..n} A000040(i). - _Jonathan Vos Post_, Jul 17 2008
%F a(A051838(n)) = A116536(n) * A007504(A051838(n)). - _Reinhard Zumkeller_, Oct 03 2011
%F A000005(a(n)) = 2^n. - _Carlos Eduardo Olivieri_, Jun 16 2015
%F a(n) = A035345(n) - A005235(n) for n > 0. - _Jonathan Sondow_, Dec 02 2015
%F For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)). - _Antti Karttunen_, Aug 30 2016
%F A054841(a(n)) = A002275(n). - _Michael De Vlieger_, Aug 31 2016
%F 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 Sum_{n>=1} 1/a(n) = A064648. - _Amiram Eldar_, Oct 16 2020
%F Sum_{n>=1} (-1)^(n+1)/a(n) = A132120. - _Amiram Eldar_, Apr 12 2021
%e 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 := n -> mul(ithprime(i),i=1..n);
%t FoldList[Times, 1, Prime[Range[20]]]
%t primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial,20] (* _José María Grau Ribas_, Feb 15 2010 *)
%t Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* _Harvey P. Dale_, Apr 11 2012 *)
%o (Haskell)
%o a002110 n = product $ take n a000040_list
%o a002110_list = scanl (*) 1 a000040_list
%o -- _Reinhard Zumkeller_, Feb 19 2012, May 03 2011
%o (Magma) [1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // _Bruno Berselli_, Oct 24 2012
%o (Magma) [1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // _Bruno Berselli_, Feb 08 2015
%o (PARI) a(n)=prod(i=1,n, prime(i)) \\ _Washington Bomfim_, Sep 23 2008
%o (PARI) p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) ) \\ _Harry J. Smith_, Nov 13 2009
%o (PARI) a(n) = factorback(primes(n)) \\ _David A. Corneth_, May 06 2018
%o (Python)
%o from sympy import primorial
%o def a(n): return 1 if n < 1 else primorial(n)
%o [a(n) for n in range(51)] # _Indranil Ghosh_, Mar 29 2017
%o (Sage) [sloane.A002110(n) for n in (1..20)] # _Giuseppe Coppoletta_, Dec 05 2014
%o (Scheme) ; with memoization-macro definec
%o (definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; _Antti Karttunen_, Aug 30 2016
%Y A034386 gives the second version of the primorial numbers.
%Y Subsequence of A005117 and of A064807. Apart from the first term, a subsequence of A083207.
%Y 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 Cf. A061720 (first differences), A143293 (partial sums).
%Y Cf. also A276085, A276086.
%Y 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 nonn,easy,nice,core
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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