A391828 Numbers k such that the map k -> k' reaches a nonzero cycle, where k' is the arithmetic derivative.

4, 27, 3125, 823543, 1647082, 2238771, 3358143, 3793738, 4425686, 4452758, 5682974, 6716282, 7251855, 8322027, 12974315, 14415850, 16650815, 18968665, 20182246, 21005098, 22838035, 24924635, 25650001, 25901246, 29126923, 30979753, 31169257, 36426306, 37322805

Offset

1,1

Comments

All numbers p^p for primes p are in the sequence, as (p^p)' = p^p.

All known terms not of the form p^p differentiate to p^p after a finite number of steps, meaning all known cycles are of length 1. It is not known if this pattern continues for every term.

A099309 is a supersequence.

A348329, excluding 0 and 1, is a subsequence.

If n = 4*k for k > 1, then n' = 4'*k+4*k' = 4*(k+k') which is a larger multiple of 4. Hence any multiple of 4 that exceeds 4 (or multiple of p^p that exceeds p^p by identical logic) diverges to infinity under the map k -> k'.

The program provided assumes that all k that diverge to infinity become a multiple of 4 after a finite number of derivatives. If false the program will not halt for some n, but any outputted values will still be correct.

Links

Table of n, a(n) for n=1..29.

Example

7251855' = 4425686
4425686' = 2238771
2238771' = 823543
823543' = 823543
Convergence to 823543 so 7251855 is in the sequence.

Prog

(Python)
def isok(n):
    if n == 0:
        return False
    derivs = [n]
    while True:
        n = A003415(n)
        if n in derivs:
            return True
        if n % 4 == 0:
            return False
        derivs.append(n)

Crossrefs

Cf. A003415, A051674, A099309, A348329.

Sequence in context: A249105 A249110 A348329 * A051674 A132641 A008973

Adjacent sequences: A391825 A391826 A391827 * A391829 A391830 A391831

Keyword

nonn

Author

Michael Adams, Dec 20 2025

Extensions

More terms from Sean A. Irvine, Jan 07 2026

Status

approved