A348425 Squares whose second arithmetic derivative is a square.

0, 1, 4, 49, 529, 2209, 6241, 27889, 28561, 35344, 49729, 128881, 192721, 250000, 431649, 528529, 703921, 1181569, 1495729, 1610361, 1868689, 3411409, 4870849, 5755201, 9138529, 11390625, 12250000, 13830961, 13845841, 15737089, 22648081, 25391521, 31618129

Offset: 1

Author

Marius A. Burtea, Oct 18 2021

Keywords

• nonn

Comments

For prime numbers of the form p = k^2 - 2 (A028871) the number m = p^2 is a term because m'' = (p^2)'' = (2*p*p')' = (2*p)'= p + 2*p' = p + 2 = k^2.

If m is a term in A028873 then p = m^2 - 3 is prime and k = p^4 is a term. Indeed: k' = 4*p^3 and k'' = 4*p^3 + 12*p^2 = 4*p^2*(p + 3) = 4*p^2*m^2.

If m is a term in A201787 then p = 5*m^2 - 6 is prime and k = p^6 is a term. Indeed: k' = 6*p^5 and k'' = 5*p^5 + 30*p^4 = 5*p^4*(p + 6) = 25*p^4*m^2.

Examples

			4'' = 4' = 4 so 4 is a term.
49'' = 14' = 9 so 49 is a term.
		

Crossrefs

Cf. A000290, A003415, A028871, A028873, A068346, A201787, A266890, A192192, A328244.

Programs

Magma

f:=func; [n:n in [s*s:s in [0.. 5623]] | IsSquare(Floor(f(Floor(f(n)))))];

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 6000]^2, IntegerQ @ Sqrt[d[d[#]]] &] (* Amiram Eldar, Oct 18 2021 *)

PARI

ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
lista(nn) = {for (n=0, nn, if (issquare(ad(ad(n^2))), print1(n^2, ", ")); ); } \\ Michel Marcus, Oct 30 2021