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A006794
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Primorial -1 primes: primes p such that -1 + product of primes up to p is prime.
(Formerly M2474)
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25
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3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113
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OFFSET
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1,1
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COMMENTS
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Or, p such that primorial(p) - 1 is prime.
Conjecture: if p# - 1 is a prime number, then the previous prime is greater than p# - exp(1)*p. - Arkadiusz Wesolowski, Jun 19 2016
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REFERENCES
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H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 111.
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LINKS
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FORMULA
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MATHEMATICA
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primorial[p_] := Product[Prime[k], {k, 1, PrimePi[p]}]; Select[Prime[Range[1900]], PrimeQ[primorial[#] - 1] &] (* Jean-François Alcover, Mar 16 2011 *)
Transpose[With[{pr=Prime[Range[2000]]}, Select[Thread[{Rest[FoldList[ Times, 1, pr]], pr}], PrimeQ[ First[#]-1]&]]][[2]] (* Harvey P. Dale, Jun 21 2011 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] - 1, _?PrimeQ]]]]] (* Eric W. Weisstein, Nov 03 2015 *)
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PROG
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(PARI) is(n)=isprime(n) && ispseudoprime(prod(i=1, primepi(n), prime(i))-1) \\ Charles R Greathouse IV, Apr 29 2015
(Python)
from sympy import nextprime, isprime
while p < 10**5:
if isprime(q-1):
p = nextprime(p)
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CROSSREFS
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Cf. A057704 (Primorial - 1 prime indices: integers n such that the n-th primorial minus 1 is prime).
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KEYWORD
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nonn,hard,more,nice,changed
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AUTHOR
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EXTENSIONS
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Stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 101; corrected in 2nd printing.
Corrected by Arlin Anderson (starship1(AT)gmail.com), who reports that he and Don Robinson have checked this sequence through about 63000 digits without finding another term (Jul 04 2000).
a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700000 have been tested)
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STATUS
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approved
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