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A260815 a(2) = 3; for n >= 3, a(n) = a(n-1) + gcd(n, a(n-1))^2.

%I A260815 #30 Sep 08 2022 08:46:13
%S A260815 3,12,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,575,
%T A260815 576,577,578,579,580,1421,1422,1423,1424,1425,1426,1427,1428,1429,
%U A260815 1430,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612
%N A260815 a(2) = 3; for n >= 3, a(n) = a(n-1) + gcd(n, a(n-1))^2.
%C A260815 The first differences of a(n) are all squares.
%H A260815 Robert Israel, <a href="/A260815/b260815.txt">Table of n, a(n) for n = 2..10000</a>
%e A260815 a(3) = 3 + gcd(3, 3)^2 = 3 + 9 = 12.
%e A260815 a(4) = 12 + gcd(4, 12)^2 = 12 + 16 = 28.
%e A260815 a(5) = 28 + gcd(5, 28)^2 = 28 + 1 = 29.
%p A260815 N:= 100: # for a(2)..a(N)
%p A260815 A:= Array(2..N):
%p A260815 A[2]:= 3:
%p A260815 for n from 3 to N do
%p A260815   A[n]:= A[n-1]+igcd(n,A[n-1])^2
%p A260815 od:
%p A260815 seq(A[i],i=2..N); # _Robert Israel_, Apr 13 2021
%t A260815 Nest[Append[#1, #1[[-1]] + GCD[#2, #1[[-1]]]^2] & @@ {#, Length[#] + 2} &, {3}, 50] (* _Michael De Vlieger_, Apr 13 2021 *)
%o A260815 (Magma) I:=[0, 3]; Remove([n le 2 select I[n] else Self(n-1)+Gcd(n, Self(n-1))^2: n in [1..52]], 1);
%Y A260815 Cf. A167195, A343402.
%K A260815 nonn,look
%O A260815 2,1
%A A260815 _Arkadiusz Wesolowski_, Jul 31 2015