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A005867
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a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).
(Formerly M1880)
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46
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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)
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OFFSET
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0,3
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COMMENTS
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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
Comment from Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 16 2006: "Let p=prime(n) and let p# be the primorial (A002110), then it can 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."
Contribution 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 [From Ben Thurston (benpaulthurston(AT)gmail.com), Aug 23 2010]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..99
Walter Nissen, Home Page (listed in lieu of email address)
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.
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FORMULA
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phi(product of first n primes), where phi = A000010.
Prod_{k=1..n} prime(k)-1.
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EXAMPLE
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a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.
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MAPLE
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a:= n-> mul(-1+ithprime(j), j=1..n): seq(a(n), n=0..18); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
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MATHEMATICA
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Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
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PROG
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(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
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CROSSREFS
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Cf. A002110.
Cf. A006093, A058254, A055768, A101301.
Sequence in context: A052667 A006925 A185135 * A192411 A179563 A079802
Adjacent sequences: A005864 A005865 A005866 * A005868 A005869 A005870
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Changed offset to 0, changed name, edited comments and examples -- T. D. Noe, Apr 04 2010
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STATUS
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approved
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