A377987 - OEIS

A377987

Irregular triangle giving on row n all those antiderivatives k of the n-th factorial, for which bigomega(k) > 2.

20, 116, 716, 2512, 5036, 40316, 84672, 176364, 1390500, 1782108, 3628773, 3628796, 10529953, 12258673, 76944384, 5338541473, 8944397353, 11690698969, 1236868096, 1849666112, 3096111708, 1004929973233, 54465962625, 1657198101073, 6791831913289, 1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209

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OFFSET

4,1

COMMENTS

Row n lists in ascending order all numbers k whose arithmetic derivative k' [A003415(k)] is equal to the n-th factorial, n! = A000142(n), and that have more than two prime factors with multiplicity, i.e., A001222(k) > 2. Rows of length zero are simply omitted, i.e., when A377986(n) = 0.

Of the initial 32 terms, 16 are odd, and of those 16 odd terms, 11 are squarefree. There are only odd terms on rows 14 and 15, why?

LINKS

Table of n, a(n) for n=4..35.

Index entries for sequences related to factorial numbers

EXAMPLE

Row n k such that A003415(k) = n! and A001222(k) > 2.

(no solutions for n = 1..3)

4: 20; (20 = 2*2*5, so 20' = 4'*5 + 5'*4 = 4*5 + 1*4 = 24 = 4!)

5: 116; (116 = 2*2*29, so 116' = 4*29 + 1*4 = 120 = 5!)

6: 716; (716 = 2*2*179, so 716' = 4*179 + 1*4 = 720 = 6!)

7: 2512, 5036;

8: 40316;