For given p = A000043(n), the following k yield a prime R and an associated (primitive) weird number W = 2^(k1)*(2^p1)*R in A258882 c A002975 c A006037:
For p = 2, no k yields a prime R = 2^k1+(2^k2)/(2^(pk)1).
For p = 3, k = 2 yields R = 5 and the (smallest) weird number W = 70 = A006037(1).
For p = 5, k = 4 yields R = 29 = a(3) and W = 7192 = A258882(3).
For p = 7, k = 4 yields R = 17 = a(2) and W = 17272 = A258882(7),
and k = 5 yields R = 41 = a(4) and W = 83312 = A258882(9).
For p = 13, k = 11 yields R = 2729 = a(5) and W = 22889716736 = A258882(288)
For p = 17, k = 13 yields R = 8737 = a(6) and W = 4690605371392 = A258882(1203).
For p = 19, k = 16 yields R = 74897 = a(8), W = 1286718208049152 = A258882(7154),
and k = 17 yields R = 174761 = a(9), W = 6004730783793152 = A258882(11466).
For p = 31, k = 16 yields R = 65537 = a(7) (smaller than both R's for p = 19),
and k = 29 yields R = 715827881 = a(10).
For p = 61, only k = 57 yields a prime R = 153722867280912929 = a(11).
For p = 89, only k = 78 yields a prime R = 302379100949042568368129 = a(12).
For p = 107 through p = 86243, no k yields a prime R.
