A284666 - OEIS

%I #38 Apr 07 2017 14:31:06

%S 1,1,1,1,25,49,18,25,32,1,841,1681,49,169,289,50,169,288,49,289,529,

%T 128,289,450,98,625,1152,289,625,961,800,841,882,162,1681,3200,288,

%U 1369,2450,529,1369,2209,1,28561,57121,49,5329,10609,961,1681,2401,289,2809,5329

%N List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

%C This is a 3-column table read by rows a, a+d, a+2*d. Each row has product a square. The rows are ordered by the products. The square roots of the products form A284876, which contains A046176. The pairs a,d form A284874.

%C Goldbach proved that a product of 3 consecutive positive integers is never a square.

%C Euler proved that a product of 4 consecutive positive integers is never a square.

%C Erdos and Selfridge (1975) proved that a product of 2 or more consecutive positive integers is never a square or a higher power.

%C Saradha (1998) proved that 18, 25, 32 is the only arithmetic progression a, a+d, ..., a+(k-1)*d whose product is a square if a>=1, 1<d<=22, and k>=3 with gcd(a,d)=1. In 1997 she showed that the product is not a square or a higher power if a>=1, 1<d<=6, and k>=3 with gcd(a,d)=1.

%C (1, 1+d, 1+2*d) is in the table if and only if d is in A078522. - _Robert Israel_, Apr 05 2017 - _Jonathan Sondow_, Apr 06 2017

%H Giovanni Resta, <a href="/A284666/b284666.txt">Table of n, a(n) for n = 1..1248</a> (triples with product < 10^18)

%H P. Erdős and J.L. Selfridge, <a href="https://projecteuclid.org/download/pdf_1/euclid.ijm/1256050816">The product of consecutive integers is never a power</a>, Illinois J. Math., 19 (1975), 292-301.

%H N. Saradha, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa82/aa8224.pdf">On perfect powers in products with terms from arithmetic progressions</a>, Acta Arith., 82 (1997), 147-172.

%H N. Saradha, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8613.pdf">Squares in products with terms in an arithmetic progression</a>, Acta Arith., 86 (1998), 27-43.

%F a(3*k+1) = A284874(2*k+1) and a(3*k+2) = A284874(2*k+1)+A284874(2*k+2) and a(3*k+3) = A284874(2*k+1)+2*A284874(2*k+2) and a(3*k+1)*a(3*k+2)*a(3*k+3) = A284876(k+1)^2 for k>=0.

%e 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2 and gcd(18,25,32) = 1, so 18,25,32 is in the sequence.

%p N:= 10^11: # to get all triples where the product <= N

%p Res:= [1,0]:

%p for a from 1 to floor(N^(1/3)) do

%p   for d from 1 while a*(a+d)*(a+2*d) <= N do

%p      if igcd(a,d) = 1 and issqr(a*(a+d)*(a+2*d)) then

%p        Res:= Res, [a,d]

%p      fi

%p   od

%p od:

%p Res:= sort([Res], (s,t) -> s[1]*(s[1]+s[2])*(s[1]+2*s[2]) <= t[1]*(t[1]+t[2])*(t[1]+2*t[2])):

%p map(t -> (t[1],t[1]+t[2],t[1]+2*t[2]), Res); # _Robert Israel_, Apr 05 2017

%t nn = 50000; t = {};

%t p[a_, b_, c_] := a *b*c; Do[

%t If[p[a, a + d, a + 2 d] <= 2 nn^2 && GCD[a, d] == 1 &&

%t    IntegerQ[Sqrt[p[a, a + d, a + 2 d]]],

%t   AppendTo[t, {a, a + d, a + 2 d}]], {a, 1, nn}, {d, 0, nn}];

%t Sort[t, p[#1[[1]], #1[[2]], #1[[3]]] <

%t     p[#2[[1]], #2[[2]], #2[[3]]] &] // Flatten

%Y Cf. A046176, A078522, A284874, A284876.

%K nonn,tabf

%O 1,5

%A _Jonathan Sondow_, Mar 31 2017