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 A290250 #34 Feb 16 2025 08:33:49
%S A290250 1248829,13,1091,13,41,37,463,13,23,13,1667,37,23,13,41,13,139
%N A290250 Smallest (prime) number a(n) > 2 such that Sum_{k=1..a(n)} k!^(2*n) is divisible by a(n).
%C A290250 If a(i) exists, then the number of primes in the sequence {Sum_{k=1..n} k!^(2*i)}_n is finite. This follows since all subsequent terms in the sum involve adding (1*2*...*a(i)*...)^(2*i) to the previous term, both of which are divisible by a(i).
%C A290250 The terms from a(19) to a(36) are 46147, 13, 587, 13, 107, 23, 41, 13, 163, 13, 43, 37, 23, 13, 397, 13, 23, 433, and the terms from a(38) to a(50) are 13, 419, 13, 9199, 23, 2129, 13, 41, 13, 2358661, 37, 409, 13. If they exist, a(18) > 25*10^6 and a(37) > 14*10^6. - _Giovanni Resta_, Jul 27 2017
%C A290250 a(37) = 17424871; a(18) > 5*10^7 - _Mark Rodenkirch_, Sep 04 2017
%H A290250 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>
%e A290250 sum(k=1..1248829, k!^2) = 14+ million-digit number which is divisible by 1248829
%e A290250 sum(k=1..13, k!^4) = 1503614384819523432725006336630745933089, which is divisible by 13
%e A290250 sum(k=1..1091, k!^6) = 17055-digit number which is divisible by 1091
%t A290250 Table[Module[{sum = 1, fac = 1, k = 2}, While[! Divisible[sum += (fac *= k)^(2 n), k], k++]; k], {n, 17}]
%Y A290250 Cf. A100289 (n such that Sum_{k=1..n} k!^2 is prime), A289945 (k!^4), A289946 (k!^6), A290014 (k!^10).
%K A290250 nonn,more,hard
%O A290250 1,1
%A A290250 _Eric W. Weisstein_, Jul 24 2017