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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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98
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1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000
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OFFSET
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0,3
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COMMENTS
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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
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
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
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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).
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LINKS
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FORMULA
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a(n) = Product_{k=1..n} (prime(k)-1) = Product_{k=1..n} A006093(n).
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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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mul(ithprime(j)-1, j=1..n) ;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Offset changed to 0, Name changed, and Comments and Examples sections edited by T. D. Noe, Apr 04 2010
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STATUS
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approved
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