A361078 Numbers k for which k = gcd(k', k"), where k' is the arithmetic derivative of k (A003415) and k" is the second derivative of k (A068346).

4, 16, 27, 64, 108, 432, 729, 1024, 2916, 3125, 4096, 6912, 12500, 16384, 19683, 27648, 46656, 50000, 84375, 110592, 186624, 314928, 337500, 746496, 800000, 823543, 1048576, 1259712, 2125764, 2278125, 3200000, 3294172, 4194304, 5038848, 5400000, 7077888, 8503056

Offset: 1

Author

Marius A. Burtea, Mar 01 2023

Keywords

• nonn

Comments

The sequence is infinite because for p prime, m = p^p (A051674) is a term.

For the prime number p, the number m = 4^p is a term. Indeed: (4^p)' = p*4^p, (4^p)" = (1 + p^2)*4^p and gcd((4^p)', 4^p) = gcd(p*4^p, (1 + p^2)*4^p) = 4^p*gcd(p, 1 + p^2) = 4^p.

Numbers of the form a*b with a, b in A051674 are terms. Indeed, if m = a*b then m' = a'*b + a*b' = a*a + b*b = 2*a*b = 2*m, m" = a*b + 2*a'b + 2*a*b' = a*b + 2*a*b + 2*a*b = 5*a*b = 5*m and gcd(m', m") = (2*m, 5*m) = m.

Examples

			4' = 4, 4" = 4 and gcd(4', 4") = gcd(4, 4) = 4, so 4 is a term.
16' = 32, 16" = 32' = 80 and gcd(16', 16") = gcd(32, 80) = 16, so 16 is a term.
		

Crossrefs

Cf. A003415 (n'), A068346 (n"), A051674, A072873, A348284.

Programs

Magma

f:=func; [n:n in [2..100000]|not IsPrime(n) and  Gcd(Floor(f(n)),Floor(f(Floor(f(n))))) eq n];

Mathematica

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^6], GCD[d[#], d[d[#]]] == # &] (* Amiram Eldar, Mar 03 2023 *)

PARI

ader(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = gcd(ader(k), ader(ader(k))) == k; \\ Michel Marcus, Mar 03 2023