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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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41
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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; internal format)
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OFFSET
| 0,3
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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 (rgwv(AT)rgwv.com), 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 (qntmpkt(AT)yahoo.com), 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
| 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
| T. D. Noe, Table of n, a(n) for n=1..100
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.
T. D. Noe, Table of n, a(n) for n=0..99
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FORMULA
| phi(product of first n primes), where phi = A000010.
Prod_{k=1..n} prime(k)-1.
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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.
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MAPLE
| with (numtheory):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
| Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
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PROG
| (PARI) for(n=0, 22, print(prod(k=1, n, prime(k)-1)))
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CROSSREFS
| Cf. A002110.
Sequence in context: A112541 A052667 A006925 * A192411 A179563 A079802
Adjacent sequences: A005864 A005865 A005866 * A005868 A005869 A005870
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Changed offset to 0, changed name, edited comments and examples -- T. D. Noe (noe(AT)sspectra.com), Apr 04 2010
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