A371164 - OEIS

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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.

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

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..47.

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

Adjacent sequences: A371161 A371162 A371163 * A371165 A371166 A371167

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 2024

STATUS

approved