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A348425 Squares whose second arithmetic derivative is a square.

%I A348425 #21 Sep 08 2022 08:46:26
%S A348425 0,1,4,49,529,2209,6241,27889,28561,35344,49729,128881,192721,250000,
%T A348425 431649,528529,703921,1181569,1495729,1610361,1868689,3411409,4870849,
%U A348425 5755201,9138529,11390625,12250000,13830961,13845841,15737089,22648081,25391521,31618129
%N A348425 Squares whose second arithmetic derivative is a square.
%C A348425 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.
%C A348425 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.
%C A348425 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.
%e A348425 4'' = 4' = 4 so 4 is a term.
%e A348425 49'' = 14' = 9 so 49 is a term.
%t A348425 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 *)
%o A348425 (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)))))];
%o A348425 (PARI) ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
%o A348425 lista(nn) = {for (n=0, nn, if (issquare(ad(ad(n^2))), print1(n^2, ", ")); ); } \\ _Michel Marcus_, Oct 30 2021
%Y A348425 Cf. A000290, A003415, A028871, A028873, A068346, A201787, A266890, A192192, A328244.
%K A348425 nonn
%O A348425 1,3
%A A348425 _Marius A. Burtea_, Oct 18 2021