A189560 Least odd number k such that x' = k has n solutions, where x' is the arithmetic derivative (A003415) of x.

3, 5, 21, 75, 151, 371, 671, 791, 311, 551, 1271, 1391, 1031, 2471, 2231, 4271, 1991, 3191, 5351, 7871, 7751, 7031, 8951, 8711, 11831, 5591, 19631, 10391, 15791, 20711, 30071, 17111, 30551, 27191, 40031, 31391, 52631, 49271, 35591, 42311, 50951, 92231

Offset: 0

Author

T. D. Noe, Apr 24 2011

Keywords

• nonn

Comments

See A189559 for k restricted to prime numbers and A189558 for no restrictions on k.

References

• See A003415.

Links

• Donovan Johnson, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A003415, A189558, A189559.

Programs

Python

from itertools import count
from sympy import factorint
def A189560(n):
    if n == 0:
        return 3
    mdict = {}
    for k in count(1,2):
        c = 0
        for m in range(1,(k**2>>2)+1):
            if m not in mdict:
                mdict[m] = sum((m*e//p for p,e in factorint(m).items()))
            if mdict[m] == k:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Sep 12 2022

Formula

a(n) is the least odd k such that A099302(k) = n.