A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

0, 1, 4, 13, 40, 186, 952, 5533, 38719, 346207, 3130816, 34444968, 382437431, 4637235152

Offset

2,3

Comments

For 1! = 1, there are an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the starting offset is 2.

Like with A351029, also here most of the solutions seem to be squarefree semiprimes, counted by A062311.

Terms a(12)..a(15) were obtained by summing the corresponding terms of A062311 and A377986.

Links

Table of n, a(n) for n=2..15.

Index entries for sequences related to factorial numbers

Formula

a(n) = A099302(A000142(n)).

a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n!], where [ ] is the Iverson bracket.

a(n) = A062311(n) + A377986(n).

Prog

(PARI)
\\ 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); };
(PARI)
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)<pv[#pv]), 0, if(!(x%s) && isprime(x/s), (x/s), 1)); };

Crossrefs

Cf. A000142, A002620, A003415, A062311, A099302, A377986.

Cf. A011371, A054861, A027868, A054896.

Cf. also A351029, A369239.

Sequence in context: A149424 A097112 A222270 * A351892 A213496 A220926

Adjacent sequences: A376407 A376408 A376409 * A376411 A376412 A376413

Keyword

nonn,hard,more

Author

Antti Karttunen, Nov 06 2024

Status

approved