This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382392 #7 Mar 24 2025 15:14:57
%S A382392 2,2,5,19,97,601,4327,35281,322571,3265949,36288017,439084817,
%T A382392 5748019201,80951270459,1220496076831,19615115520037,334764638208037,
%U A382392 6046686277632071,115242726703104073,2311256907767808001,48658040163532800037,1072909785605898240031
%N A382392 a(n) is the least prime number whose factorial base expansion contains the digit n.
%C A382392 This sequence is well defined: a(0) = a(1) = 2, and for n > 1, (n+1)! and n*n! + 1 are coprime, so by Dirichlet's theorem on arithmetic progressions, there exists a prime number p of the form k*(n+1)! + n*n! + 1 for some k >= 0, and the factorial base expansion of this prime number contains the digit n, hence a(n) <= p.
%H A382392 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A382392 a(n) > A001563(n).
%e A382392 The initial terms, in decimal and in factorial base, are:
%e A382392 n a(n) fact(a(n))
%e A382392 - ------- -----------------
%e A382392 0 2 1,0
%e A382392 1 2 1,0
%e A382392 2 5 2,1
%e A382392 3 19 3,0,1
%e A382392 4 97 4,0,0,1
%e A382392 5 601 5,0,0,0,1
%e A382392 6 4327 6,0,0,1,0,1
%e A382392 7 35281 7,0,0,0,0,0,1
%e A382392 8 322571 8,0,0,0,0,1,2,1
%e A382392 9 3265949 9,0,0,0,0,1,0,2,1
%o A382392 (PARI) a(n) = { forprime (p = n*n!, oo, my (q = p); for (r = 2, oo, if (q==0, break, q % r==n, return (p), q \= r););); }
%Y A382392 Cf. A001563, A062584, A090703.
%K A382392 nonn,base
%O A382392 0,1
%A A382392 _Rémy Sigrist_, Mar 23 2025