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

%I A348329 #24 Oct 12 2022 08:58:05
%S A348329 0,1,4,27,3125,823543,1647082,2238771
%N A348329 Numbers k such that k' = k'', where ' is the arithmetic derivative.
%C A348329 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
%C A348329 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
%F A348329 Numbers k such that A003415(k) = A068346(k).
%p A348329 isA348329 := proc(n)
%p A348329     local d ;
%p A348329     d := A003415(n) ;
%p A348329     if A003415(d) = d then
%p A348329         true ;
%p A348329     else
%p A348329         false;
%p A348329     end if;
%p A348329 end proc:
%p A348329 for n from 0 do
%p A348329     if isA348329(n) then
%p A348329         print(n) ;
%p A348329     end if;
%p A348329 end do: # _R. J. Mathar_, Oct 19 2021
%t A348329 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 *)
%o A348329 (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A348329 isok(k) = ad(k) == ad(ad(k)); \\ _Michel Marcus_, Oct 18 2021
%o A348329 (Python)
%o A348329 from sympy import factorint
%o A348329 from itertools import count, islice
%o A348329 def ad(n): return 0 if n<2 else sum(n*e//p for p, e in factorint(n).items())
%o A348329 def agen(): yield from (k for k in count(0) if (adk:=ad(k)) == ad(adk))
%o A348329 print(list(islice(agen(), 5))) # _Michael S. Branicky_, Oct 12 2022
%Y A348329 Cf. A003415, A051674, A068346, A100408.
%K A348329 nonn,more
%O A348329 1,3
%A A348329 _Wesley Ivan Hurt_, Oct 12 2021