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A371164
Rectangular array, read by antidiagonals: row k shows all numbers whose prime factorization p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k) has e(i) > 0 for i=1..k.
0
2, 4, 6, 8, 12, 30, 16, 18, 60, 210, 32, 24, 90, 420, 2310, 64, 36, 120, 630, 4620, 30030, 128, 48, 150, 840, 6930, 60060, 510510, 256, 54, 180, 1050, 9240, 90090, 1021020, 9699690, 512, 72, 240, 1260, 11550, 120120, 1531530, 19399380, 223092870, 1024, 96
OFFSET
1,1
EXAMPLE
Corner:
2 4 8 16 32 64
6 12 18 24 36 48
30 60 90 120 150 180
210 420 630 840 1050 1260
2310 4620 6930 9240 11550 13860
30030 60060 90090 120120 150150 180180
90 = 2^1 * 3^2 * 5^1, so 90 is in row 3.
MATHEMATICA
rows = 10; t = (Table[n = 120000/Exp[row] Apply[Times, Table[Prime[k], {k, row}]];
eq = Table[Prime[i]^k[i], {i, row}];
Sort[Flatten[Table[Apply[Times, eq], ##] & @@Table[{k[i],
Log[Prime[i], n/Apply[Times, Take[eq, i - 1]]]}, {i, row}]]], {row, rows}]);
Table[t[[n]][[k]], {n, 1, rows}, {k, 1, rows}] // TableForm (* array *)
w[n_, k_] := t[[n]][[k]];
Table[w[n - k + 1, k], {n, rows}, {k, n, 1, -1}] // Flatten (* sequence *)
(* Peter J. C. Moses, Mar 23 2024 *)
CROSSREFS
Cf. A000040, A055932 (union of all rows, preceded by 1).
Sequence in context: A131885 A173941 A194406 * A087443 A059901 A036035
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 26 2024
STATUS
approved