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ICF Quartus
The table shows the index-correlated factorization of the first 67 terms of the quartus, as well as the composition of the terms in terms of previous terms.
The indexing is derived from the primus in the following way:
Q2n-1 = Un + Un-1   (definition)

For brevety we'll represent the formula as Q(2n-1) = n+(n-1), for example Q3=2+1.

Note that the indices of the 'U' terms of every single factor sum to the corresponding factor of the index.

Index-correlated factorization gives a one-on-one mapping with regard to the index.
Once introduced factors reappear following the quartus' basic property Q2n-1 | Q(2k-1)(2n-1).

The occurence of prime-indices as actual factors of the corresponding term
A prime factor 'p' appears in a term on index Qp for F=(p+2) and, following the quartus' basic property, reappears in the same series on Q(2k-1)p for every natural k.
For primes bigger than 3, all other factors on prime-indices will always be 'near-multiples' of the form 2kp±1.

 Quartus Factors Q1 = (1+0) Q3 = (2+1) Q5 = (3+2) Q7 = (4+3) Q9 = (2+1) * (4-(2-1)) Q11 = (6+5) Q13 = (7+6) Q15 = (2+1) * (7-(5-4)-(2-1)) Q17 = (9+8) Q19 = (10+9) Q21 = (2+1) * (10-(8-7)-(5-4)-(2-1)) Q23 = (12+11) Q25 = (3+2) * (11-(9-6)-(4-1)) Q27 = (2+1) * (4-(2-1)) * (10-(8-1)) Q29 = (15+14) Q31 = (16+15) Q33 = (2+1) * (16-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q35 = (3+2) * (16-(14-11)-(9-6)-(4-1)) Q37 = (19+18) Q39 = (2+1) * (19-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q41 = (21+20) Q43 = (22+21) Q45 = (2+1) * (4-(2-1)) * (19-(17-10)-(8-1)) Q47 = (24+23) Q49 = (4+3) * (22-(20-15)-(13-8)-(6-1)) Q51 = (2+1) * (25-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q53 = (27+26) Q55 = (3+2) * (26-(24-21)-(19-16)-(14-11)-(9-6)-(4-1)) Q57 = (2+1) * (28-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q59 = (30+29) Q61 = (31+30) Q63 = (2+1) * (4-(2-1)) * (28-(26-19)-(17-10)-(8-1)) Q65 = (3+2) * (31-(29-26)-(24-21)-(19-16)-(14-11)-(9-6)-(4-1)) Q67 = (34+33) Q69 = (2+1) * (34-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q71 = (36+35) Q73 = (37+36) Q75 = (2+1) * (7-(5-4)-(2-1)) * (31-(29-16)-(14-1)) Q77 = (4+3) * (36-(34-29)-(27-22)-(20-15)-(13-8)-(6-1)) Q79 = (40+39) Q81 = (2+1) * (4-(2-1)) * (10-(8-1)) * (28-(26-1)) Q83 = (42+41) Q85 = (3+2) * (41-(39-36)-(34-31)-(29-26)-(24-21)-(19-16)-(14-11)-(9-6)-(4-1)) Q87 = (2+1) * (43-(41-40)-(38-37)-(35-34)-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q89 = (45+44) Q91 = (4+3) * (43-(41-36)-(34-29)-(27-22)-(20-15)-(13-8)-(6-1)) Q93 = (2+1) * (46-(44-43)-(41-40)-(38-37)-(35-34)-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q95 = (3+2) * (46-(44-41)-(39-36)-(34-31)-(29-26)-(24-21)-(19-16)-(14-11)-(9-6)-(4-1)) Q97 = (49+48) Q99 = (2+1) * (4-(2-1)) * (46-(44-37)-(35-28)-(21-19)-(17-10)-(8-1)) Q101 = (51+50) Q103 = (52+51) Q105 = (2+1) * (7-(5-4)-(2-1)) * (46-(44-31)-(29-16)-(14-1)) Q107 = (54+53) Q109 = (55+54) Q111 = (2+1) * (55-(53-52)-(50-49)-(47-46)-(44-43)-(41-40)-(38-37)-(35-34)-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q113 = (57+56) Q115 = (3+2) * (56-(54-51)-(49-46)-(44-41)-(39-36)-(34-31)-(29-26)-(24-21)-(19-16)-(14-11)-(9-6)-(4-1)) Q117 = (2+1) * (4-(2-1)) * (55-(53-46)-(44-37)-(35-28)-(21-19)-(17-10)-(8-1)) Q119 = (4+3) * (57-(55-50)-(48-43)-(41-36)-(34-29)-(27-22)-(20-15)-(13-8)-(6-1)) Q121 = (6+5) * (56-(54-45)-(43-34)-(32-23)-(21-12)-(10-1)) Q123 = (2+1) * (61-(59-58)-(56-55)-(53-52)-(50-49)-(47-46)-(44-43)-(41-40)-(38-37)-(35-34)-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q125 = (3+2) * (11-(9-6)-(4-1)) * (51-(49-26)-(24-1)) Q127 = (64+63) Q129 = (2+1) * (64-(62-61)-(59-58)-(56-55)-(53-52)-(50-49)-(47-46)-(44-43)-(41-40)-(38-37)-(35-34)-(32-31)-(29-28)-(26-25)-(23-22)-(20-19)-(17-16)-(14-13)-(11-10)-(8-7)-(5-4)-(2-1)) Q131 = (66+65) Q133 = (4+3) * (64-(62-57)-(55-50)-(48-43)-(41-36)-(34-29)-(27-22)-(20-15)-(13-8)-(6-1))

You may be interested in the method involved.