A099302 -id:A099302 - cp's OEIS frontend

A377993 Number of integers whose arithmetic derivative (A003415) is equal to A024451(n), the arithmetic derivative of the n-th primorial.

1, 2, 3, 4, 6, 3, 2, 330, 3, 3

Offset: 2

Antti Karttunen, Nov 20 2024

nonn

a(n) is the number of natural numbers k such that k' = A003415(k) = A024451(n). The solutions k are listed in A377992.

For 1! = 1, there is an infinite number of integers k for which k' = 1 (all the primes), therefore the starting offset is 2.

			There is just one number such that A003415(k) = A024451(2) = 5, and that is k=6, therefore a(2) = 1.
There are two numbers such that A003415(k) = A024451(3) = 31, and they are k=30 and k=58, therefore a(3) = 2.

Index entries for sequences related to primorial numbers

Row lengths of A377992.

Cf. A002620, A003415, A024451, A099302.

a(n) = A099302(A024451(n)).

a(n) = Sum_{k=1..A002620(A024451(n))} [A003415(k) = A024451(n)], where [ ] is the Iverson bracket.

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

Original entry on oeis.org

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

Craig J. Beisel, Sep 09 2022

nonn

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

			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).

E. J. Barbeau, Remarks on an arithmetic derivative, Canadian Mathematical Bulletin, Volume 4, Issue 2, May 1961, pp. 117-122.
Wikipedia, Arithmetic_derivative.

Cf. A003415.

Bisection of A099302.

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", "); );

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

a(n) = A099302(2n).

cp's OEIS Frontend

A377993 Number of integers whose arithmetic derivative (A003415) is equal to A024451(n), the arithmetic derivative of the n-th primorial.

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