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

Offset: 4

Author

Antti Karttunen, Nov 21 2024

Keywords

• nonn
• tabf
• hard
• more

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?

Examples

			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;
    9:   84672, 176364; (2^6 * 3^3 * 7^2 and 2^2 * 3^3 * 23 * 71)
   10:   1390500, 1782108, 3628773, 3628796, 10529953, 12258673;
   11:   (no solutions)
   12:   76944384, 5338541473, 8944397353, 11690698969;
   13:   1236868096, 1849666112, 3096111708, 1004929973233;
   14:   54465962625, 1657198101073, 6791831913289;
   15:   1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209;
   etc.
Note that although A003415(9) = 6 = 3!, it is not included in this table as 9 is a semiprime, with A001222(9) = 2.
		

Links

• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A001222, A003415, A377986 (row lengths).

Cf. also A366890, A369240, A377992.

Programs

PARI

\\ Use the programs given in A377987 and A376410.
\\ the data needs also to be post-processed (sorted) with
\\ sols = sort_solutions_vector(readvec("a_terms_for_A377987_unsorted.txt"));
\\ using these functions:
sort_solutions_vector(v) = vecsort(v,sort_by_A003415_and_magnitude);
sort_by_A003415_and_magnitude(x,y) = { my(s = sign(A003415(x)-A003415(y))); if(!s, sign(x-y), s); };