Primes of the form Sum[ Sum[ Prime[k], {k,1,m} ], {m,1,n} ] = A014148[n].

Terms

a(0) =2a(1) =7a(2) =17a(3) =103a(4) =467a(5) =6577a(6) =17189a(7) =61627a(8) =109919a(9) =130531a(10) =198109a(11) =239579a(12) =399557a(13) =559313a(14) =1291909a(15) =1807609a(16) =2851951a(17) =3644479a(18) =5189339a(19) =5659651a(20) =6794993a(21) =6905357a(22) =7359091a(23) =9795109a(24) =19405363a(25) =20319427a(26) =36052151a(27) =40343053a(28) =42229639a(29) =44970689