A377987 - OEIS

%I #16 Nov 21 2024 15:30:25

%S 20,116,716,2512,5036,40316,84672,176364,1390500,1782108,3628773,

%T 3628796,10529953,12258673,76944384,5338541473,8944397353,11690698969,

%U 1236868096,1849666112,3096111708,1004929973233,54465962625,1657198101073,6791831913289,1307674367996,5739085040351,21522396453889,63577408859233,104747513922049,287711613106993,626768279186209

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

%C 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.

%C 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?

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%e     (no solutions for n = 1..3)

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

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

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

%e     7:   2512, 5036;

%e     8:   40316;

%e     9:   84672, 176364; (2^6 * 3^3 * 7^2 and 2^2 * 3^3 * 23 * 71)

%e    10:   1390500, 1782108, 3628773, 3628796, 10529953, 12258673;

%e    11:   (no solutions)

%e    12:   76944384, 5338541473, 8944397353, 11690698969;

%e    13:   1236868096, 1849666112, 3096111708, 1004929973233;

%e    14:   54465962625, 1657198101073, 6791831913289;

%e    15:   1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209;

%e    etc.

%e Note that although A003415(9) = 6 = 3!, it is not included in this table as 9 is a semiprime, with A001222(9) = 2.

%o (PARI)

%o \\ Use the programs given in A377987 and A376410.

%o \\ the data needs also to be post-processed (sorted) with

%o \\ sols = sort_solutions_vector(readvec("a_terms_for_A377987_unsorted.txt"));

%o \\ using these functions:

%o sort_solutions_vector(v) = vecsort(v,sort_by_A003415_and_magnitude);

%o sort_by_A003415_and_magnitude(x,y) = { my(s = sign(A003415(x)-A003415(y))); if(!s, sign(x-y), s); };

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

%Y Cf. also A366890, A369240, A377992.

%K nonn,tabf,hard,more

%O 4,1

%A _Antti Karttunen_, Nov 21 2024