A348329 Numbers k such that k' = k'', where ' is the arithmetic derivative.

0, 1, 4, 27, 3125, 823543, 1647082, 2238771

Offset: 1

Author

Wesley Ivan Hurt, Oct 12 2021

Keywords

• nonn
• more

Comments

For n > 2, a(n) is such that a(n)' = p^p for some prime p. So A051674 is a subsequence. - David A. Corneth, Oct 13 2021

If p > 2 and p^p-2 are both primes (i.e., p is an odd prime term of A100408), then 2*(p^p-2) is a term. Terms of this type are 1647082 and 3956839311320627178247954, corresponding to p = 7 and 19 respectively. - Amiram Eldar, Oct 13 2021

Crossrefs

Cf. A003415, A051674, A068346, A100408.

Programs

Maple

isA348329 := proc(n)
    local d ;
    d := A003415(n) ;
    if A003415(d) = d then
        true ;
    else
        false;
    end if;
end proc:
for n from 0 do
    if isA348329(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Oct 19 2021

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 2.5*10^6], d[#] == d[d[#]] &] (* Amiram Eldar, Oct 13 2021 *)

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = ad(k) == ad(ad(k)); \\ Michel Marcus, Oct 18 2021

Python

from sympy import factorint
from itertools import count, islice
def ad(n): return 0 if n<2 else sum(n*e//p for p, e in factorint(n).items())
def agen(): yield from (k for k in count(0) if (adk:=ad(k)) == ad(adk))
print(list(islice(agen(), 5))) # Michael S. Branicky, Oct 12 2022

Formula

Numbers k such that A003415(k) = A068346(k).