OFFSET
1,3
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.
EXAMPLE
4'' = 4' = 4 so 4 is a term.
49'' = 14' = 9 so 49 is a term.
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 *)
PROG
(Magma) f:=func<n | n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [n:n in [s*s:s in [0.. 5623]] | IsSquare(Floor(f(Floor(f(n)))))];
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Oct 18 2021
STATUS
approved