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Numbers k such that !k * k! + 1 = A003422(k) * A000142(k) + 1 = A061640(k) + 1 is prime.
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%I #13 Oct 03 2024 23:34:56

%S 1,2,4,6,13,14,21,41,64,110,268,1196,3083,5201,7496,9698

%N Numbers k such that !k * k! + 1 = A003422(k) * A000142(k) + 1 = A061640(k) + 1 is prime.

%C a(1)-a(11) were found by Earls (2003-2004).

%C The corresponding primes are 2, 5, 241, 110881, 3256459844769331201, ... .

%H Jason Earls, <a href="https://www.proquest.com/openview/4375c07dc57d66a121a0bd7bcf5f86d5/1?pq-origsite=gscholar&amp;cbl=31970">Marching Primes: !N x N! + 1</a>, Journal of Recreational Mathematics, Vol. 32, No. 2 (2003-2004), pp. 123-124.

%t Select[Range[300], PrimeQ[#! * Sum[k!, {k, 0, #-1}] + 1] &]

%o (PARI) is(k) = isprime(k! * sum(i = 0, k-1, i!) + 1);

%Y Cf. A000142, A003422, A061640.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Oct 01 2024

%E a(14)-a(16) from _Michael S. Branicky_, Oct 03 2024