A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1). (Formerly M1880)

1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000

Offset

0,3

Comments

Local minima of Euler's phi function. - Walter Nissen

Number of potential primes in a modulus primorial(n+1) sieve. - Robert G. Wilson v, Nov 20 2000

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

From Gary W. Adamson, Apr 21 2009: (Start)

Equals (-1)^n * (1, 1, 1, 2, 8, 48, ...) dot (-1, 2, -3, 5, -7, 11, ...).

a(6) = 480 = (1, 1, 1, 2, 8, 48) dot (-1, 2, -3, 5, -7, 11) = (-1, 2, -3, 10, -56, 528). (End)

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

First column of A096294. - Eric Desbiaux, Jun 20 2013

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

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

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

For n > 1, a(n) is the number of prime(n+1)-rough numbers in [1, primorial(prime(n))]. - Alexandre Herrera, Aug 29 2023

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..99

Larry Deering, The Black Key Sieve, Box 275, Bellport NY 11713-0275, 1998.

Alphonse de Polignac, Six propositions arithmologiques déduites du crible d'Ératosthène, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 1, Tome 8 (1849), pp. 423-429. See p. 425.

Frank Ellermann, Illustration for A002110, A005867, A038110, A060753

Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.

Dennis Martin, Proofs Regarding Primorial Patterns [via Internet Archive Wayback-machine]

Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author]

F. E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991

Travis Near, Improving MATLAB's isprime performance without arbitrary-precision arithmetic, arXiv:2108.04791 [cs.MS], 2021.

John K. Sellers, Distribution of twin primes in repeating sequences of prime factors, arXiv:2108.00288 [math.GM], 2021. See Table 1 p. 11.

Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

Formula

a(n) = phi(product of first n primes) = A000010(A002110(n)).

a(n) = Product_{k=1..n} (prime(k)-1) = Product_{k=1..n} A006093(n).

Sum_{n>=0} a(n)/A002110(n+1) = 1. - Bob Selcoe, Jan 09 2015

a(n) = A002110(n)-((1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1)). - Jamie Morken, Mar 27 2019

a(n) = |Sum_{k=0..n} A070918(n,k)|. - Alois P. Heinz, Aug 18 2019

a(n) = A058251(n)/A060753(n+1). - Jamie Morken, Apr 25 2022

Example

a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.

Maple

A005867 := proc(n)
    mul(ithprime(j)-1, j=1..n) ;
end proc: # Zerinvary Lajos, Aug 24 2008, R. J. Mathar, May 03 2017

Mathematica

Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
RecurrenceTable[{a[0]==1, a[n]==(Prime[n]-1)a[n-1]}, a, {n, 20}] (* Harvey P. Dale, Dec 09 2013 *)
EulerPhi@ FoldList[Times, 1, Prime@ Range@ 18] (* Michael De Vlieger, Mar 18 2016 *)

Prog

(PARI) for(n=0, 22, print1(prod(k=1, n, prime(k)-1), ", "))
(Haskell)
a005867 n = a005867_list !! n
a005867_list = scanl (*) 1 a006093_list
-- Reinhard Zumkeller, May 01 2013

Crossrefs

Cf. A000010, A002110, A006093, A054640, A058254, A055768, A070918, A101301, A058251, A060753.

Cf. A057773 (2-adic valuation).

Column 1 of A281890.

Sequence in context: A006925 A185135 A238805 * A280133 A192411 A179563

Adjacent sequences: A005864 A005865 A005866 * A005868 A005869 A005870

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Offset changed to 0, Name changed, and Comments and Examples sections edited by T. D. Noe, Apr 04 2010

Status

approved