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

1,5

Comments

Conjecture: All terms are positive with the exception of a(1).

Links

Table of n, a(n) for n=1..100.

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

Cf. A003415.

Bisection of A099302.

Sequence in context: A242879 A176775 A175778 * A226182 A099774 A305973

Adjacent sequences: A357036 A357037 A357038 * A357040 A357041 A357042

Keyword

nonn

Author

Craig J. Beisel, Sep 09 2022

Status

approved