A377986
Number of integers k, with bigomega(k) > 2, whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.
Original entry on oeis.org
0, 0, 0, 1, 1, 1, 2, 1, 2, 6, 0, 4, 4, 3, 7
Offset: 1
See the examples in A377987.
Row lengths of irregular triangle
A377987.
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A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A377986(n) = { my(g=n!); sum(k=1,A002620(g),(bigomega(k)>2) && (A003415(k)==g)); };
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A377986(n) = AntiDeriv(n!,2,"a_terms_for_A377987_unsorted.txt"); \\ The rest of the program is given in A376410.
A377987
Irregular triangle giving on row n all those antiderivatives k of the n-th factorial, for which bigomega(k) > 2.
Original entry on oeis.org
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
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.
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\\ 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); };
Showing 1-2 of 2 results.
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