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Integer solutions of the arithmetic differential equation m' = m - sqrt(m).
2

%I #18 Jun 12 2022 22:52:03

%S 1,9,441,815409,553613841

%N Integer solutions of the arithmetic differential equation m' = m - sqrt(m).

%C a(6) > 10^17. - _Giovanni Resta_, Mar 20 2014

%C m = k^2, where k satisfies k' = (k-1)/2. - _Charlie Neder_, Mar 08 2019

%e For m = 441 we have that m' = 420, sqrt(441) = 21 and 420 = 441 - 21.

%p with(numtheory); P:= proc(q) local n,p,x;

%p for n from 1 to q do x:=n^2;

%p if x*add(op(2,p)/op(1,p),p=ifactors(x)[2])=n^2-n then print(n^2);

%p fi; od; end: P(10^9);

%Y Cf. A003415, A239478.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Mar 20 2014