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 A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#. (Formerly M1691 N0668) 1530
 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090 (list; graph; refs; listen; history; text; internal format)
 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 n-th 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 x-th root of the x-th 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 "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 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 Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50. 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). D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74. 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], 2011-2012; Ramanujan J., 29 (2012), 359-384. Harvey Dubner, Factorial and primorial primes, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197-203. (Annotated scanned copy) F. Ellermann, Illustration for A002110, A005867, A038110, A060753 S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210. D. J. Greenhoe, MRA-Wavelet 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 linear-time algorithm for the orbit problem over cyclic groups, preprint, CONCUR 2014 - Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp 327-341. A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, CONCUR 2014 - Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp 327-341. R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. Thomas Morrill, Further Development of "Non-Pythagorean" 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), 375-388. Patrick Sole and Michel Planat, The Robin inequality for 7-free 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 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). - Juri-Stepan 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]] primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial, 20] (* José María Grau Ribas, Feb 15 2010 *) Join[{1}, Denominator[Accumulate[1/Prime[Range]]]] (* 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)  cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012 (Magma)  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 memoization-macro 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 STATUS approved

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Last modified September 26 08:43 EDT 2022. Contains 356993 sequences. (Running on oeis4.)