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 A037153 a(n) = p-n!, where p is the smallest prime > n!+1. 19
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 The first 4003 terms are prime. - Dana Jacobsen, May 10 2015 LINKS Ray Chandler and Dana Jacobsen, Table of n, a(n) for n = 1..4000 [first 1200 terms from Ray Chandler] Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4. Hisanori Mishima, Primes near to factorial, Dec 2008. MATHEMATICA 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 *) PROG (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 (PARI) a(n)=nextprime(n!+2)-n! \\ Charles R Greathouse IV, Jul 02 2013; Corrected by Dana Jacobsen, May 10 2015 (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 print([a(n) for n in range(1, 66)]) # Michael S. Branicky, May 22 2022 CROSSREFS Cf. A087202, A005235, A033932. Sequence in context: A111060 A082432 A336298 * A323185 A168065 A077724 Adjacent sequences:  A037150 A037151 A037152 * A037154 A037155 A037156 KEYWORD nonn AUTHOR EXTENSIONS Edited by N. J. A. Sloane, Mar 06 2010 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)