cp's OEIS Frontend

A365850 Numbers k for which k^2 + (k')^2 is a square, where k' is the arithmetic derivative of k (A003415).

%I A365850 #21 Nov 18 2023 13:11:55
%S A365850 0,1,12,15,35,81,143,323,400,441,899,1540,1763,2700,3599,4641,5183,
%T A365850 5929,9375,10395,10403,11663,13585,18225,19043,21952,22499,30576,
%U A365850 32399,35581,36863,39203,48841,51983,57599,72899,79523,97343,121103,148176,166375,175692,176399
%N A365850 Numbers k for which k^2 + (k')^2 is a square, where k' is the arithmetic derivative of k (A003415).
%C A365850 If p and p + 2 are twin primes (A001359) then m = p*(p + 2) is a term. Indeed, m' = p + (p + 2) = 2*p + 2 and m^2 + (m')^2 = p^2*(p + 2)^2 + (2*p + 2)^2 = (p^2 + 2*p + 2)^2.
%C A365850 More generally, if p and p + 2*k, k >= 1, are prime numbers, then m = p^k*(p +  2*k)^k is a term. Indeed, m' = k*p^(k - 1)*(p + 2*k)^k+ k*p^k*(p + 2*k)^(k - 1) = k*p^(k - 1)*(p + 2*k)^(k-1)*(2*p + 2*k). Thus, m^2 + (m')^2 = p^(2*k)*(p +  2*k)^(2*k) + (k^2)*p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*p + 2*k)^2 = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(p^2*(p + 2*k)^2 + k^2*(2*p + 2*k)^2) = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*k^2 + 2*k*p + p^2)^2.
%e A365850 For k = 12, k' = 16 and 12^2 + 16^2 = 144 + 256 = 400 = 20^2, so 12 is a term.
%e A365850 For k = 15, k' = 8 and 15^2 + 8^2 = 225 + 64 = 289 = 17^2, so 15 is a term.
%e A365850 For k = 143, k' = 24 and 143^2 + 24^2 = 144 + 256 = 21025 = 145^2, so 143 is a term.
%p A365850 ader:= proc(n) local f;   n*add(f[2]/f[1], f=ifactors(n)[2]) end proc:
%p A365850 select(t -> issqr(t^2 + ader(t)^2), [$0..10^6]; # _Robert Israel_, Oct 17 2023
%t A365850 d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 180000], IntegerQ[Sqrt[#^2 + d[#]^2]] &] (* _Amiram Eldar_, Oct 11 2023 *)
%o A365850 (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 [0..200000] |IsSquare( n^2+( Floor(f(n))^2))];
%Y A365850 Cf. A077800, A153417 - A153419, A156104, A242476, A252089 - A252091, A271347, A271981, A271982, A272176.
%K A365850 nonn
%O A365850 1,3
%A A365850 _Marius A. Burtea_, Oct 09 2023