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a(n) is the quotient of the sum of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n), and prime(n).
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%I #14 Jan 29 2023 19:30:06

%S 0,1,7,139,365641,39916801,1317933016441,355688356705921,

%T 53128667010491295649,10888872347627347035630931201,

%U 8841761993746245283777145088001,10333147966386144929666651337523200000001

%N a(n) is the quotient of the sum of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n), and prime(n).

%C a(n) is prime for n = {3,4,5,6,7,31,738}; a(738) ~ 7.1 * 10^18518. There are no others for n up to 1000. - _Peter J. C. Moses_, Mar 03 2014

%e Let n=4, prime(n)=7. Consider integers i!+6!/i!, i=1,2,3,4,5: 721,362,126,54,126. Among them 721,126,126 are divisible by 7. So a(4)=(721 + 126 + 126)/7 = 139.

%t Map[Total[Cases[Table[i!+(#-1)!/i!,{i,#-2}]/#,_Integer]]&,Prime[Range[10]]] (* _Peter J. C. Moses_, Mar 10 2014 *)

%Y Cf. A238444, A007619.

%K nonn

%O 1,3

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Mar 03 2014