cp's OEIS Frontend

A357039 Number of integer solutions to x' = 2n, where x' is the arithmetic derivative of x.

%I A357039 #20 Sep 12 2022 22:49:11
%S A357039 0,1,1,1,2,2,2,3,2,2,3,4,3,2,3,4,4,4,2,3,4,4,4,6,4,3,5,4,4,7,3,5,6,3,
%T A357039 5,7,5,5,7,6,5,8,5,4,9,6,5,8,3,6,8,5,6,9,6,8,10,6,6,13,4,6,10,4,7,9,6,
%U A357039 5,8,9,8,11,6,5,12,5,8,12,5,8,11,6,6,14,9,6,11,9,7,14,6,8,13,7,8,13,7,9,13,8
%N A357039 Number of integer solutions to x' = 2n, where x' is the arithmetic derivative of x.
%C A357039 Conjecture: All terms are positive with the exception of a(1).
%H A357039 E. J. Barbeau, <a href="https://doi.org/10.4153/CMB-1961-013-0">Remarks on an arithmetic derivative</a>, Canadian Mathematical Bulletin, Volume 4, Issue 2, May 1961, pp. 117-122.
%H A357039 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_derivative">Arithmetic_derivative</a>.
%F A357039 a(n) = A099302(2n).
%e A357039 Since 12'=16, 39'=16 and 55'=16, a(8)=3. We don't need to search any higher than (x'^2)/4=(16^2)/4=64 from Barbeau lower bound (See links).
%o A357039 (PARI) for(n=1, 100, v=2*n; c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); print1(c", "); );
%o A357039 (Python)
%o A357039 from sympy import factorint
%o A357039 def A357039(n): return sum(1 for m in range(1,n**2+1) if sum((m*e//p for p,e in factorint(m).items())) == n<<1) # _Chai Wah Wu_, Sep 12 2022
%Y A357039 Cf. A003415.
%Y A357039 Bisection of A099302.
%K A357039 nonn
%O A357039 1,5
%A A357039 _Craig J. Beisel_, Sep 09 2022