Partial-Sum -> Some Primes

[Leroy Quet]

Let a(1) = 1;

Let a(m) be the lowest yet unpicked positive integer such that:

sum{k=1 to m} k* a(k)

is a prime.

Figured by hand, so very well could be wrong, I get:

a(k) : 1, 2, 4, 3, 6, 5, 12, 7,...

Question:
Is this a permutation of the positive integers?

If so, its inverse-permutation is:

1, 2, 4, 3, 6, 5, 8,...

(I get some EIS matches for the few terms I give for the inverse 
permutation, but no matches for the primary sequence.)

thanks,