Factorizations from first 1000 terms of https://oeis.org/A046376/b046376.txt that do not contain 1.

[2, 3]
[2, 30203]
[2, 30403]
[2, 32323]
[2, 32423]
[3, 30203]
[3, 32323]
[2, 3002003]
[2, 3222223]
[2, 3223223]
[2, 3233323]
[2, 3304033]
[2, 3343433]
[2, 3400043]
[2, 3424243]
[2, 3443443]
[2, 3444443]
[3, 3002003]
[3, 3222223]
[3, 3223223]
[3, 3233323]
[2, 300020003]
[2, 302030203]
[2, 302202203]
[2, 302303203]
[2, 302333203]
[2, 302343203]
[2, 302444203]
[2, 304242403]
[2, 304404403]
[2, 304444403]
[2, 320444023]
[2, 322030223]
[2, 322222223]
[2, 322323223]
[2, 323222323]
[2, 323232323]
[2, 323333323]
[2, 323343323]
[2, 324202423]
[2, 324232423]
[2, 324343423]
[2, 324434423]
[2, 330404033]
[2, 330424033]
[2, 332000233]
[2, 332424233]
[2, 333040333]
[2, 333434333]
[2, 340444043]
[2, 342020243]
[2, 342202243]
[2, 342232243]
[2, 342323243]
[2, 343020343]
[2, 344040443]
[3, 300020003]
[3, 302030203]
[3, 302202203]
[3, 302303203]
[3, 302333203]
[3, 322030223]
[3, 322222223]
[3, 322323223]
[3, 323222323]
[3, 323232323]
[3, 323333323]
[3, 332000233]