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Search: id:A005235
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| A005235 |
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Fortunate numbers: least m>1 such that m+prime(n)# is prime, where p# denotes the product of the primes <= p.
(Formerly M2418)
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+0
33
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| 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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R. F. Fortune conjectured that a(n) is always prime.
a(n) is the smallest m such that m > 1 and A002110(n)+m is prime. For every n, a(n) must be greater than prime(n+1)-1. - Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Aug 20 2003
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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).
Martin Gardner, The Last Recreations (1997), pp. 194-95.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly, 95 (1988), 697-712.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..2000 (a(n)+prime(n)# is a probable prime)
C. Banderier, Conjecture checked for n<1000 [It has been reported that that the data given here contains several errors]
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Author?, McEachen Conjecture
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FORMULA
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If x(n) = 1 + product(prime(i), i=1..n), q(n) = least prime > x(n), then a(n) = q(n)-x(n)+1.
a(n) = 1 + the difference between the n-th Primorial plus one and the next prime.
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EXAMPLE
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a(4) = 13 because P_4! = 2*3*5*7 = 210, plus one is 211, the next prime is 223 and the difference between 210 and 223 is 13.
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MATHEMATICA
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NextPrime[ n_Integer ] := Module[ {k}, k = n + 1; While[ ! PrimeQ[ k ], k++ ]; k ]; Fortunate[ n_Integer ] := Module[ {p, q}, p = Product[ Prime[ i ], {i, 1, n} ] + 1; q = NextPrime[ p ]; q - p + 1 ]; Table[ Fortunate[ n ], {n, 1, 60} ]
r[n_] := (For[m=(Prime[n+1]+1)/2, !PrimeQ[Product[Prime[k], {k, n}]+2m-1], m++ ]; 2m-1); Table[a[n], {n, 60}]
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CROSSREFS
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Cf. A046066, A002110, A006862, A035345, A035346, A055211.
Cf. A129912.
Sequence in context: A154700 A051507 A060274 this_sequence A107664 A085013 A164939
Adjacent sequences: A005232 A005233 A005234 this_sequence A005236 A005237 A005238
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Jud McCranie, j.mccranie(AT)comcast.net.
Guy lists 100 terms, as computed by Stan Wagon.
The first 500 terms are primes - Robert G. Wilson v
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