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

945, 1575, 2205, 7425, 78975, 131625, 342225, 570375, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 16695, 18585, 19215, 21105, 22365, 22995, 24885, 26145, 28035, 28215, 29835

Offset: 3

Author

T. D. Noe, Mar 31 2011

Keywords

• nonn
• tabf

Comments

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.

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.]

Examples

			From _M. F. Hasler_, Jul 27 2016: (Start)
Row 3: 945, 1575, 2205, 7425, 78975, 131625, 342225, 570375;
Row 4: 3465, 4095, 5355, ...(571 more)..., 249450402403828125, 970969744245403125;
Row 5: 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, ...
Row 6: 692835, 838695, 937365, 1057485, 1130415, 1181895, 1225785, 1263405, ...
Row 7: 22309287, 28129101, 30069039, 34051017, 35888853, 36399363, ...
The first column is A188342 = (945, 3465, 15015, 692835, 22309287, ...) (End)
		

Links

• T. D. Noe, Rows n = 3..4, flattened
• L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
• A. Ferrier, Table errata 176, MTAC 4 (1950), 222.
• Fritz Herzog, Table Errata 571, Math. Comp. 34 (1980), 652.
• T. D. Noe, 576 odd primitive abundant numbers, factored

Crossrefs

Row lengths are A303933.

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).