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A037155 a(n) = n!-p, where p is the largest prime < (n!-1).

%I A037155 #31 Jul 02 2025 16:01:56
%S A037155 3,5,7,11,17,31,13,11,13,13,23,17,47,53,59,41,101,31,31,73,89,73,149,
%T A037155 37,43,101,31,79,61,163,47,193,113,127,97,79,73,83,131,79,109,109,53,
%U A037155 89,79,103,59,97,179,67,59,127,61,461,277,109,137,139,71,71,101,359
%N A037155 a(n) = n!-p, where p is the largest prime < (n!-1).
%C A037155 Analogous to the lesser Fortunate numbers and like them, all entries so far are primes.
%H A037155 Charles R Greathouse IV, <a href="/A037155/b037155.txt">Table of n, a(n) for n = 3..1000</a>
%H A037155 Antonín Čejchan, Michal Křížek, and Lawrence Somer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Krizek/krizek3.html">On Remarkable Properties of Primes Near Factorials and Primorials</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
%F A037155 a(n) >= n. - _Seiichi Manyama_, Mar 21 2018
%e A037155 a(4) = 4!-19 = 24-19 = 5.
%t A037155 PrevPrime[ n_Integer ] := (k=n-1; While[ !PrimeQ[ k ], k-- ]; Return[ k ]); f[ n_Integer ] := (p = n! - 1; q = NextPrime[ p ]; Return[ p - q + 1 ]); Table[ f[ n ], {n, 3, 75} ]
%t A037155 f[n_]:=Module[{nf=n!},nf-NextPrime[nf-1,-1]];f/@Range[3,90]  (* _Harvey P. Dale_, Mar 20 2011 *)
%o A037155 (PARI) a(n)=my(N=n!); N-precprime(N-3) \\ _Charles R Greathouse IV_, Jan 28 2018
%o A037155 (Python)
%o A037155 from sympy import factorial, prevprime
%o A037155 def a(n): fn = factorial(n); return fn - prevprime(fn-1)
%o A037155 print([a(n) for n in range(3, 65)]) # _Michael S. Branicky_, May 22 2022
%Y A037155 Cf. A055211.
%K A037155 nonn
%O A037155 3,1
%A A037155 _Jud McCranie_
%E A037155 More terms from _James Sellers_, Jul 06 2000