A357039 Number of integer solutions to x' = 2n, where x' is the arithmetic derivative of x.
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, 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, 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
Offset: 1
Keywords
Examples
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).
Links
- E. J. Barbeau, Remarks on an arithmetic derivative, Canadian Mathematical Bulletin, Volume 4, Issue 2, May 1961, pp. 117-122.
- Wikipedia, Arithmetic_derivative.
Programs
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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", "); );
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Python
from sympy import factorint 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
Formula
a(n) = A099302(2n).
Comments