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A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).
(Formerly M1880)
46
1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000 (list; graph; refs; listen; history; text; internal format)
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 "About this sequence" 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(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

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

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.

F. Ellermann, Illustration for A002110, A005867, A038110, A060753

Dennis Martin, About this sequence

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

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

a:= n-> mul(-1+ithprime(j), j=1..n): seq(a(n), n=0..18); # Zerinvary Lajos, Aug 24 2008

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 *)

PROG

(PARI) for(n=0, 22, print(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. A002110, A006093, A058254, A055768, A101301.

Sequence in context: A006925 A185135 A238805 * A192411 A179563 A079802

Adjacent sequences:  A005864 A005865 A005866 * A005868 A005869 A005870

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Changed offset to 0, changed name, edited comments and examples by T. D. Noe, Apr 04 2010

STATUS

approved

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Last modified October 31 03:18 EDT 2014. Contains 248845 sequences.