cp's OEIS Frontend

A377987 Irregular triangle giving on row n all those antiderivatives k of the n-th factorial, for which bigomega(k) > 2.

%I A377987 #16 Nov 21 2024 15:30:25
%S A377987 20,116,716,2512,5036,40316,84672,176364,1390500,1782108,3628773,
%T A377987 3628796,10529953,12258673,76944384,5338541473,8944397353,11690698969,
%U A377987 1236868096,1849666112,3096111708,1004929973233,54465962625,1657198101073,6791831913289,1307674367996,5739085040351,21522396453889,63577408859233,104747513922049,287711613106993,626768279186209
%N A377987 Irregular triangle giving on row n all those antiderivatives k of the n-th factorial, for which bigomega(k) > 2.
%C A377987 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 A377987 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 A377987 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%e A377987 Row n    k such that A003415(k) = n! and A001222(k) > 2.
%e A377987     (no solutions for n = 1..3)
%e A377987     4:   20;   (20 = 2*2*5, so 20' = 4'*5 + 5'*4 = 4*5 + 1*4 = 24 = 4!)
%e A377987     5:   116;  (116 = 2*2*29, so 116' = 4*29 + 1*4 = 120 = 5!)
%e A377987     6:   716;  (716 = 2*2*179, so 716' = 4*179 + 1*4 = 720 = 6!)
%e A377987     7:   2512, 5036;
%e A377987     8:   40316;
%e A377987     9:   84672, 176364; (2^6 * 3^3 * 7^2 and 2^2 * 3^3 * 23 * 71)
%e A377987    10:   1390500, 1782108, 3628773, 3628796, 10529953, 12258673;
%e A377987    11:   (no solutions)
%e A377987    12:   76944384, 5338541473, 8944397353, 11690698969;
%e A377987    13:   1236868096, 1849666112, 3096111708, 1004929973233;
%e A377987    14:   54465962625, 1657198101073, 6791831913289;
%e A377987    15:   1307674367996, 5739085040351, 21522396453889, 63577408859233, 104747513922049, 287711613106993, 626768279186209;
%e A377987    etc.
%e A377987 Note that although A003415(9) = 6 = 3!, it is not included in this table as 9 is a semiprime, with A001222(9) = 2.
%o A377987 (PARI)
%o A377987 \\ Use the programs given in A377987 and A376410.
%o A377987 \\ the data needs also to be post-processed (sorted) with
%o A377987 \\ sols = sort_solutions_vector(readvec("a_terms_for_A377987_unsorted.txt"));
%o A377987 \\ using these functions:
%o A377987 sort_solutions_vector(v) = vecsort(v,sort_by_A003415_and_magnitude);
%o A377987 sort_by_A003415_and_magnitude(x,y) = { my(s = sign(A003415(x)-A003415(y))); if(!s, sign(x-y), s); };
%Y A377987 Cf. A000142, A001222, A003415, A377986 (row lengths).
%Y A377987 Cf. also A366890, A369240, A377992.
%K A377987 nonn,tabf,hard,more
%O A377987 4,1
%A A377987 _Antti Karttunen_, Nov 21 2024