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A037153
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a(n) = p-n!, where p is the smallest prime > n!+1.
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19
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2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 47, 67, 223, 107, 127, 79, 37, 97, 61, 131, 311, 43, 97, 53, 61, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271, 67, 193
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OFFSET
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1,1
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COMMENTS
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Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 1200 terms are primes. Are all terms prime?
a(n) is the first (smallest) m such that m > 1 and n!+ m is prime. The second such m is A087202(n). a(n) must be greater than nextprime(n)-1. - Farideh Firoozbakht, Sep 01 2003
Sequence A069941, which counts the primes between n! and n!+n^2, provides numerical evidence that the smallest prime p greater than n!+1 is a prime distance from n!; that is, p-n! is a prime number. For p-n! to be a composite number, p would have to be greater than n!+n^2, which would imply that A069941(n)=0. - T. D. Noe, Mar 06 2010
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LINKS
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MATHEMATICA
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NextPrime[ n_Integer ] := (k=n+1; While[ !PrimeQ[ k ], k++ ]; Return[ k ]); f[ n_Integer ] := (p = n! + 1; q = NextPrime[ p ]; Return[ q - p + 1 ]); Table[ f[ n ], {n, 1, 75} ] (* Robert G. Wilson v *)
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PROG
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(Magma) z:=125; [p-f where p is NextPrime(f+1) where f is Factorial(n): n in [1..z]]; // Klaus Brockhaus, Mar 02 2010
(MuPAD) for n from 1 to 65 do f := n!:a := nextprime(f+2)-f:print(a) end_for; // Zerinvary Lajos, Feb 22 2007
(Perl) use ntheory ":all"; for my $n (1..1000) { my $f=factorial($n); say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 10 2015
(Python)
from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(fn+1) - fn
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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