A034758 Dirichlet convolution of primes (with 1) with Bell numbers.

1, 3, 5, 12, 22, 70, 216, 909, 4165, 21207, 116004, 678741, 4213634, 27644897, 190899424, 1382960388, 10480142200, 82864878340, 682076806220, 5832742247551, 51724158236078, 474869816388803, 4506715738447402