cp's OEIS Frontend

A352330 Squares whose arithmetic derivative (A003415) is a cube.

%I A352330 #21 Apr 01 2022 23:47:45
%S A352330 0,1,11664,20736,2313441,2985984,9150625,28005264,236421376,655360000,
%T A352330 1871773696,3340840000,4294967296,10435031104,10485760000,11716114081,
%U A352330 33556377856,50054665441,80706559921,156531800881,203928109056,258439040161,282429536481,414998793616
%N A352330 Squares whose arithmetic derivative (A003415) is a cube.
%C A352330 For p prime number of the form p = 4*m^3 - 1 (A199367) the number k = 2^8*p^4 is a term. Indeed, k' = (2^8*p^4)' = 8*2^7*p^4 + 2^8*4*p^3 = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*(2*(p + 1)) = 2^9*p^3*2*4*m^3 = (2^3*p*8*m)^3 so k is a term.
%C A352330 The sequence is infinite because numbers of the form m = 2^(2^(6*k + 5)), k >= 0, are terms. Indeed: m' = 2^(6*k + 5)*2^(2^(6*k + 5) - 1) = 2^(6*k + 4 + 2^(6*k + 5)) = 2^(6*k + 3 + 2^(6*k + 5) + 1), and the exponent 6*k + 3 + 2^(6*k + 5) + 1 is divisible by 3.
%C A352330 If p is a prime number then the numbers of the form m = p^(64^k), k >= 1 are terms.
%e A352330 11664 = 108^2 and 11664' = 46656 = 36^3 so 11664 is a term.
%e A352330 20736 = 144^2 and 20376' = 110592 = 48^3 so 20736 is a term.
%t A352330 d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6.5*10^5]^2, IntegerQ@Surd[d[#], 3] &] (* _Amiram Eldar_, Mar 13 2022 *)
%o A352330 (Magma) f:=func<n|n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]:i in [1..#Factorisation(n)]])>; [p:p in [s*s:s in [0.. 450000]]| IsPower(Floor(f(p)),3)];
%Y A352330 Cf. A003415, A199367, A266890.
%K A352330 nonn
%O A352330 1,3
%A A352330 _Marius A. Burtea_, Mar 13 2022