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A284876 Positive integers that are square roots of products a*(a+d)*(a+2*d) with coprime a > 0, d >= 0.

%I A284876 #39 Apr 07 2017 14:31:18
%S A284876 1,35,120,1189,1547,1560,2737,4080,8400,13175,24360,29520,31080,39997,
%T A284876 40391,52633,62279,65773,80520,93023,131040,133055,133560,185640,
%U A284876 212219,240240,241345,379680,385440,393805,399960,434231,449497,471240,510229,555360,585395
%N A284876 Positive integers that are square roots of products a*(a+d)*(a+2*d) with coprime a > 0, d >= 0.
%C A284876 The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The pairs a, d form A284874.
%C A284876 sqrt((1+d)*(1+2*d)) is a member if and only if d is in A078522. The values of sqrt((1+d)*(1+2*d)) form the subsequence A046176.
%H A284876 Giovanni Resta, <a href="/A284876/b284876.txt">Table of n, a(n) for n = 1..416</a> (terms < 10^9)
%F A284876 a(k+1)^2 = A284666(3*k+1)*A284666(3*k+2)*A284666(3*k+3) = A284874(2*k+1)*(A284874(2*k+1) + A284874(2*k+2))*(A284874(2*k+1) + 2*A284874(2*k+2)) for k >= 0.
%e A284876 gcd(1,24)=1 and 1*(1+24)*(1+2*24) = 25*49 = (5*7)^2, so 5*7 = 35 is a member.
%e A284876 gcd(18,7)=1 and 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2, so 3*5*8 = 120 is in the sequence.
%t A284876 nn = 50000; t = {};
%t A284876 p[a_, b_, c_] := a b c; Do[
%t A284876 If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&
%t A284876    IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],
%t A284876   AppendTo[t, Sqrt[p[a, a + d, a + 2 d]]]], {a, 1, nn}, {d, 0, nn}]; Sort[t]
%o A284876 (PARI) is(n,s)={!fordiv(n*=n,a,a^3>n && return;issquare(n\a*8+a^2,&s) && (s-=3*a)%4==0 && gcd(s\4,a)==1 && break)} \\ _M. F. Hasler_, Apr 06 2017
%Y A284876 Cf. A046176, A078522, A284666, A284874.
%K A284876 nonn
%O A284876 1,2
%A A284876 _Jonathan Sondow_, Apr 05 2017
%E A284876 a(19)-a(37) from _Giovanni Resta_, Apr 06 2017