

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
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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

Table of n, a(n) for n=1..17.
Eric Weisstein's World of Mathematics, Factorial Sums


EXAMPLE

sum(k=1..1248829, k!^2) = 14+ milliondigit number which is divisible by 1248829
sum(k=1..13, k!^4) = 1503614384819523432725006336630745933089, which is divisible by 13
sum(k=1..1091, k!^6) = 17055digit 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



