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A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).

%I A005867 M1880 #263 Aug 09 2025 16:23:17
%S A005867 1,1,2,8,48,480,5760,92160,1658880,36495360,1021870080,30656102400,
%T A005867 1103619686400,44144787456000,1854081073152000,85287729364992000,
%U A005867 4434961926979584000,257227791764815872000,15433667505888952320000
%N A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).
%C A005867 Local minima of Euler's phi function. - _Walter Nissen_
%C A005867 Number of potential primes in a modulus primorial(n+1) sieve. - _Robert G. Wilson v_, Nov 20 2000
%C A005867 Let p=prime(n) and let p# be the primorial (A002110), then it can be shown that any p# consecutive numbers have exactly a(n-1) numbers whose lowest prime factor is p. For a proof, see the "Proofs Regarding Primorial Patterns" link. For example, if we let p=7 and consider the interval [101,310] containing 210 numbers, we find the 8 numbers 119, 133, 161, 203, 217, 259, 287, 301. - Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 16 2006
%C A005867 From _Gary W. Adamson_, Apr 21 2009: (Start)
%C A005867 Equals (-1)^n * (1, 1, 1, 2, 8, 48, ...) dot (-1, 2, -3, 5, -7, 11, ...).
%C A005867 a(6) = 480 = (1, 1, 1, 2, 8, 48) dot (-1, 2, -3, 5, -7, 11) = (-1, 2, -3, 10, -56, 528). (End)
%C A005867 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 A005867 First column of A096294. - _Eric Desbiaux_, Jun 20 2013
%C A005867 Conjecture: The g.f. for the prime(n+1)-rough numbers (A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063) is x*P(x)/(1-x-x^a(n)+x^(a(n)+1)), where P(x) is an order a(n) polynomial with symmetric coefficients (i.e., c(0)=c(n), c(1)=c(n-1), ...). - _Benedict W. J. Irwin_, Mar 18 2016
%C A005867 a(n)/A002110(n+1) (primorial(n+1)) is the ratio of natural numbers whose smallest prime factor is prime(n+1); i.e., prime(n+1) coprime to A002110(n). So the ratio of even numbers to natural numbers = 1/2; odd multiples of 3 = 1/6; multiples of 5 coprime to 6 (A084967) = 2/30 = 1/15; multiples of 7 coprime to 30 (A084968) = 8/210 = 4/105; etc. - _Bob Selcoe_, Aug 11 2016
%C A005867 The 2-adic valuation of a(n) is A057773(n), being sum of the 2-adic valuations of the product terms here. - _Kevin Ryde_, Jan 03 2023
%C A005867 For n > 1, a(n) is the number of prime(n+1)-rough numbers in [1, primorial(prime(n))]. - _Alexandre Herrera_, Aug 29 2023
%D A005867 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005867 T. D. Noe, <a href="/A005867/b005867.txt">Table of n, a(n) for n = 0..99</a>
%H A005867 Larry Deering, <a href="http://www.qsl.net/w2gl/blackkey.html">The Black Key Sieve</a>, Box 275, Bellport NY 11713-0275, 1998.
%H A005867 Alphonse de Polignac, <a href="http://www.numdam.org/item/NAM_1849_1_8__423_1/">Six propositions arithmologiques déduites du crible d'Ératosthène</a>, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 1, Tome 8 (1849), pp. 423-429. See p. 425.
%H A005867 Frank Ellermann, <a href="/A005867/a005867.txt">Illustration for A002110, A005867, A038110, A060753</a>.
%H A005867 Ken Hicks and Kevin Ward, <a href="https://arxiv.org/abs/2108.03268">Series and Product Relations Made from Primes</a>, arXiv:2108.03268 [math.NT], 2021.
%H A005867 Dennis Martin, <a href="http://web.archive.org/web/20140501151819/http://primenace.com/papers/math/PrimorialPatternProofs.htm">Proofs Regarding Primorial Patterns</a> [via Internet Archive Wayback-machine]
%H A005867 Dennis Martin, <a href="/A005867/a005867.pdf">Proofs Regarding Primorial Patterns</a> [Cached copy, with permission of the author]
%H A005867 Francis 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 A005867 Travis Near, <a href="https://arxiv.org/abs/2108.04791">Improving MATLAB's isprime performance without arbitrary-precision arithmetic</a>, arXiv:2108.04791 [cs.MS], 2021.
%H A005867 John K. Sellers, <a href="https://arxiv.org/abs/2108.00288">Distribution of twin primes in repeating sequences of prime factors</a>, arXiv:2108.00288 [math.GM], 2021. See Table 1 p. 11.
%H A005867 Andrew V. Sutherland, <a href="http://hdl.handle.net/1721.1/38881">Order Computations in Generic Groups</a>, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
%F A005867 a(n) = phi(product of first n primes) = A000010(A002110(n)).
%F A005867 a(n) = Product_{k=1..n} (prime(k)-1) = Product_{k=1..n} A006093(n).
%F A005867 Sum_{n>=0} a(n)/A002110(n+1) = 1. - _Bob Selcoe_, Jan 09 2015
%F A005867 a(n) = A002110(n)-((1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1)). - _Jamie Morken_, Mar 27 2019
%F A005867 a(n) = |Sum_{k=0..n} A070918(n,k)|. - _Alois P. Heinz_, Aug 18 2019
%F A005867 a(n) = A058251(n)/A060753(n+1). - _Jamie Morken_, Apr 25 2022
%F A005867 a(n) = A002110(n) - A016035(A002110(n)) - 1 for n >= 1. - _David James Sycamore_, Sep 07 2024
%F A005867 Sum_{n>=0} 1/a(n) = A345974. - _Amiram Eldar_, Jun 26 2025
%e A005867 a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.
%p A005867 A005867 := proc(n)
%p A005867     mul(ithprime(j)-1,j=1..n) ;
%p A005867 end proc: # _Zerinvary Lajos_, Aug 24 2008, _R. J. Mathar_, May 03 2017
%t A005867 Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
%t A005867 RecurrenceTable[{a[0]==1,a[n]==(Prime[n]-1)a[n-1]},a,{n,20}] (* _Harvey P. Dale_, Dec 09 2013 *)
%t A005867 EulerPhi@ FoldList[Times, 1, Prime@ Range@ 18] (* _Michael De Vlieger_, Mar 18 2016 *)
%o A005867 (PARI) for(n=0, 22, print1(prod(k=1,n, prime(k)-1), ", "))
%o A005867 (Haskell)
%o A005867 a005867 n = a005867_list !! n
%o A005867 a005867_list = scanl (*) 1 a006093_list
%o A005867 -- _Reinhard Zumkeller_, May 01 2013
%Y A005867 Cf. A000010, A002110, A006093, A054640, A058254, A055768, A070918, A101301, A058251, A060753.
%Y A005867 Cf. A057773 (2-adic valuation).
%Y A005867 Column 1 of A281890.
%Y A005867 Cf. A016035, A345974.
%K A005867 nonn,easy,nice
%O A005867 0,3
%A A005867 _N. J. A. Sloane_
%E A005867 Offset changed to 0, Name changed, and Comments and Examples sections edited by _T. D. Noe_, Apr 04 2010