OFFSET
1,5
COMMENTS
Conjecture: All terms are positive with the exception of a(1).
LINKS
E. J. Barbeau, Remarks on an arithmetic derivative, Canadian Mathematical Bulletin, Volume 4, Issue 2, May 1961, pp. 117-122.
Wikipedia, Arithmetic_derivative.
FORMULA
a(n) = A099302(2n).
EXAMPLE
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).
PROG
(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", "); );
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Craig J. Beisel, Sep 09 2022
STATUS
approved