factorial

1!+2!+...+n! = 2^{2k+1}+1 <==> n = 2,3,4 & k = 0,1,2

1!+2!+...+n! = 2^{2k+1}+1·4!+1 <==> n = 5 & k = 3

1!+2!+...+n! = 2^{2k+1}+3·5!+1 <==> n = 6 & k = 4