cp's OEIS Frontend

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).

%I A361078 #20 Apr 10 2023 00:09:41
%S A361078 4,16,27,64,108,432,729,1024,2916,3125,4096,6912,12500,16384,19683,
%T A361078 27648,46656,50000,84375,110592,186624,314928,337500,746496,800000,
%U A361078 823543,1048576,1259712,2125764,2278125,3200000,3294172,4194304,5038848,5400000,7077888,8503056
%N 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).
%C A361078 The sequence is infinite because for p prime, m = p^p (A051674) is a term.
%C A361078 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.
%C A361078 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.
%e A361078 4' = 4, 4" = 4 and gcd(4', 4") = gcd(4, 4) = 4, so 4 is a term.
%e A361078 16' = 32, 16" = 32' = 80 and gcd(16', 16") = gcd(32, 80) = 16, so 16 is a term.
%t A361078 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 *)
%o A361078 (Magma) f:=func<h |h le 1 select 0 else h*(&+[Factorisation(h)[i][2] / Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n:n in [2..100000]|not IsPrime(n) and  Gcd(Floor(f(n)),Floor(f(Floor(f(n))))) eq n];
%o A361078 (PARI) ader(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A361078 isok(k) = gcd(ader(k), ader(ader(k))) == k; \\ _Michel Marcus_, Mar 03 2023
%Y A361078 Cf. A003415 (n'), A068346 (n"), A051674, A072873, A348284.
%K A361078 nonn
%O A361078 1,1
%A A361078 _Marius A. Burtea_, Mar 01 2023