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

1, 35, 120, 1189, 1547, 1560, 2737, 4080, 8400, 13175, 24360, 29520, 31080, 39997, 40391, 52633, 62279, 65773, 80520, 93023, 131040, 133055, 133560, 185640, 212219, 240240, 241345, 379680, 385440, 393805, 399960, 434231, 449497, 471240, 510229, 555360, 585395

Offset: 1

Author

Jonathan Sondow, Apr 05 2017

Keywords

• nonn

Comments

The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The pairs a, d form A284874.

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.

Examples

			gcd(1,24)=1 and 1*(1+24)*(1+2*24) = 25*49 = (5*7)^2, so 5*7 = 35 is a member.
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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..416 (terms < 10^9)

Crossrefs

Cf. A046176, A078522, A284666, A284874.

Programs

Mathematica

nn = 50000; t = {};
p[a_, b_, c_] := a b c; Do[
If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&
   IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],
  AppendTo[t, Sqrt[p[a, a + d, a + 2 d]]]], {a, 1, nn}, {d, 0, nn}]; Sort[t]

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

Formula

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.

Extensions

a(19)-a(37) from Giovanni Resta, Apr 06 2017