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 A290250 Smallest (prime) number a(n) > 2 such that Sum_{k=1..a(n)} k!^(2*n) is divisible by a(n). 0
 1248829, 13, 1091, 13, 41, 37, 463, 13, 23, 13, 1667, 37, 23, 13, 41, 13, 139 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). 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 a(37) = 17424871; a(18) > 5*10^7 - Mark Rodenkirch, Sep 04 2017 LINKS Eric Weisstein's World of Mathematics, Factorial Sums EXAMPLE sum(k=1..1248829, k!^2) = 14+ million-digit number which is divisible by 1248829 sum(k=1..13, k!^4) = 1503614384819523432725006336630745933089, which is divisible by 13 sum(k=1..1091, k!^6) = 17055-digit number which is divisible by 1091 MATHEMATICA Table[Module[{sum = 1, fac = 1, k = 2}, While[! Divisible[sum += (fac *= k)^(2 n), k], k++]; k], {n, 17}] CROSSREFS Cf. A100289 (n such that Sum_{k=1..n} k!^2 is prime), A289945 (k!^4), A289946 (k!^6), A290014 (k!^10). Sequence in context: A206019 A210084 A212489 * A251156 A210132 A251118 Adjacent sequences:  A290247 A290248 A290249 * A290251 A290252 A290253 KEYWORD nonn,more,hard AUTHOR Eric W. Weisstein, Jul 24 2017 STATUS approved

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Last modified May 28 09:09 EDT 2022. Contains 354112 sequences. (Running on oeis4.)