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A188439 Irregular triangle of odd primitive abundant numbers (A006038) in which row n has numbers with n distinct prime factors.

%I A188439 #28 Oct 08 2022 14:15:35
%S A188439 945,1575,2205,7425,78975,131625,342225,570375,3465,4095,5355,5775,
%T A188439 5985,6435,6825,7245,8085,8415,8925,9135,9555,9765,11655,12705,12915,
%U A188439 13545,14805,16695,18585,19215,21105,22365,22995,24885,26145,28035,28215,29835
%N A188439 Irregular triangle of odd primitive abundant numbers (A006038) in which row n has numbers with n distinct prime factors.
%C A188439 The initial row has 8 terms. Row n begins with A188342(n). Dickson proves that each row has a finite number of terms. He lists the first two rows in factored form in his paper. However, as Ferrier and Herzog report, Dickson's tables have many errors. There are 576 odd primitive abundant numbers (OPAN) having 4 distinct prime factors, the last of which is 3^10 5^5 17^4 251^2 = 970969744245403125. The next row, for 5 distinct prime factors, has over 100000 terms.
%C A188439 If the prime factors were counted with multiplicity, then the table would start with row 5, having 121 terms: (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). Row 6 would start (7425, 28215, 29835, 33345, 34155, ...), and row 7, (81081, 121095, 164835, 182655, 189189, ...). - _M. F. Hasler_, Jul 27 2016 [See A287646.]
%H A188439 T. D. Noe, <a href="/A188439/b188439.txt">Rows n = 3..4, flattened</a>
%H A188439 L. E. Dickson, <a href="http://www.jstor.org/stable/2370405">Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors</a>, American Journal of Mathematics 35 (1913), pp. 413-422.
%H A188439 A. Ferrier, <a href="http://www.ams.org/journals/mcom/1950-04-032/S0025-5718-50-99455-5/S0025-5718-50-99455-5.pdf">Table errata 176</a>, MTAC 4 (1950), 222.
%H A188439 Fritz Herzog, <a href="http://www.ams.org/journals/mcom/1980-34-150/S0025-5718-80-99963-9/S0025-5718-80-99963-9.pdf">Table Errata 571</a>, Math. Comp. 34 (1980), 652.
%H A188439 T. D. Noe, <a href="/A188439/a188439.txt">576 odd primitive abundant numbers, factored</a>
%e A188439 From _M. F. Hasler_, Jul 27 2016: (Start)
%e A188439 Row 3: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375;
%e A188439 Row 4: 3465, 4095, 5355, ...(571 more)..., 249450402403828125, 970969744245403125;
%e A188439 Row 5: 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, ...
%e A188439 Row 6: 692835, 838695, 937365, 1057485, 1130415, 1181895, 1225785, 1263405, ...
%e A188439 Row 7: 22309287, 28129101, 30069039, 34051017, 35888853, 36399363, ...
%e A188439 The first column is A188342 = (945, 3465, 15015, 692835, 22309287, ...) (End)
%Y A188439 Row lengths are A303933.
%Y A188439 Cf. A006038 (all OPAN), A188342 (first column of this table), A287646 (variant where row n contains all OPAN with n prime factors counted with multiplicity).
%K A188439 nonn,tabf
%O A188439 3,1
%A A188439 _T. D. Noe_, Mar 31 2011