A376410
Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.
Original entry on oeis.org
0, 1, 4, 13, 40, 186, 952, 5533, 38719, 346207, 3130816, 34444968, 382437431, 4637235152
Offset: 2
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\\ Slow program, for computing just a few terms:
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]));
A376410(n) = { my(g=n!); sum(k=1,A002620(g),A003415(k)==g); };
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A376410(n) = AntiDeriv(n!);
AntiDeriv(n,startvlen=1,solsfilename="") = { my(v = vector(startvlen,i,2), ip = #v, r, c=0); while(1, r = A003415vrl(v,n); if(0==r, ip--, if(r > 1, c++; if(solsfilename!="", write(solsfilename, r*factorback(v)))); ip = #v); if(0==ip, v = vector(1+#v,i,2); ip = #v; if(A003415vec(v) > n, return(c)), v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1]))); };
A003415vec(tv) = { my(n=factorback(tv), s=0, m=1, spf); for(i=1,#tv,spf = tv[i]; n /= spf; s += m*n; m *= spf); (s); }; \\ Compute Arithmetic derivative from the vector of primes.
A003415vrl(pv,lim) = { my(n=factorback(pv), x=lim-n, s=0, m=1, spf, u=n); for(i=1,#pv,spf = pv[i]; u /= spf; s += m*u; m *= spf); if(((x/s)
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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