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

0, 1, 11664, 20736, 2313441, 2985984, 9150625, 28005264, 236421376, 655360000, 1871773696, 3340840000, 4294967296, 10435031104, 10485760000, 11716114081, 33556377856, 50054665441, 80706559921, 156531800881, 203928109056, 258439040161, 282429536481, 414998793616

Offset: 1

Author

Marius A. Burtea, Mar 13 2022

Keywords

• nonn

Comments

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.

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.

If p is a prime number then the numbers of the form m = p^(64^k), k >= 1 are terms.

Examples

			11664 = 108^2 and 11664' = 46656 = 36^3 so 11664 is a term.
20736 = 144^2 and 20376' = 110592 = 48^3 so 20736 is a term.
		

Crossrefs

Cf. A003415, A199367, A266890.

Programs

Magma

f:=func; [p:p in [s*s:s in [0.. 450000]]| IsPower(Floor(f(p)),3)];

Mathematica

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 *)