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A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.

%I A067363 #15 May 25 2023 13:44:18
%S A067363 2,3,7,5,17,11,17,23,23,103,59,17,29,79,59,23,347,307,53,227,131,83,
%T A067363 67,223,29,59,197,83,181,293,71,71,139,43,67,103,431,743,1279,197,419,
%U A067363 127,271,73,229,503,211,181,1597,151,151,197,1013,179,587,71,137,547
%N A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.
%C A067363 The first 118 terms are primes. Are all terms prime? For n!^i, with 0<i<6, it looks like the terms are prime, too (see references). But for n!^6: a(28)=1189=29*41.
%C A067363 The first 2278 terms are primes. - _Dana Jacobsen_, May 13 2015
%H A067363 Dana Jacobsen, <a href="/A067363/b067363.txt">Table of n, a(n) for n = 1..2278</a>
%H A067363 Cyril Banderier, <a href="http://algo.inria.fr/banderier/Computations/prime_factorial.html">Fortunate Numbers</a>
%t A067363 a[n_] := For[i=2, True, i++, If[PrimeQ[n!^3+i], Return[i]]]
%t A067363 spn[n_]:=Module[{c=(n!)^3},NextPrime[c+1]-c]; Array[spn,60] (* _Harvey P. Dale_, May 25 2023 *)
%o A067363 (MuPAD) for n from 1 to 50 do f := n!^3:a := nextprime(f+2)-f:print(a) end_for
%o A067363 (PARI) for(n=1, 100, f=n!^3; print1(nextprime(f+2)-f,", ")) \\ _Dana Jacobsen_, May 13 2015
%o A067363 (Perl) use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**3; say "$n ", next_prime($f+1)-$f; } # _Dana Jacobsen_, May 13 2015
%Y A067363 Cf. A037153, A037153, A005235, A067362, A067364, A067365.
%K A067363 nonn
%O A067363 1,1
%A A067363 Frank Buss (fb(AT)frank-buss.de), Jan 19 2002
%E A067363 Edited by _Dean Hickerson_, Mar 02 2002