cp's OEIS Frontend

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

%I A376410 #20 Nov 21 2024 15:30:11
%S A376410 0,1,4,13,40,186,952,5533,38719,346207,3130816,34444968,382437431,
%T A376410 4637235152
%N A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.
%C A376410 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.
%C A376410 Like with A351029, also here most of the solutions seem to be squarefree semiprimes, counted by A062311.
%C A376410 Terms a(12)..a(15) were obtained by summing the corresponding terms of A062311 and A377986.
%H A376410 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A376410 a(n) = A099302(A000142(n)).
%F A376410 a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n!], where [ ] is the Iverson bracket.
%F A376410 a(n) = A062311(n) + A377986(n).
%o A376410 (PARI)
%o A376410 \\ Slow program, for computing just a few terms:
%o A376410 A002620(n) = ((n^2)>>2);
%o A376410 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A376410 A376410(n) = { my(g=n!); sum(k=1,A002620(g),A003415(k)==g); };
%o A376410 (PARI)
%o A376410 A376410(n) = AntiDeriv(n!);
%o A376410 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]))); };
%o A376410 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.
%o A376410 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)); };
%Y A376410 Cf. A000142, A002620, A003415, A062311, A099302, A377986.
%Y A376410 Cf. A011371, A054861, A027868, A054896.
%Y A376410 Cf. also A351029, A369239.
%K A376410 nonn,hard,more
%O A376410 2,3
%A A376410 _Antti Karttunen_, Nov 06 2024