A284874 -id:A284874 - cp's OEIS frontend

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

Jonathan Sondow, Apr 05 2017

nonn

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.

			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.

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

Cf. A046176, A078522, A284666, A284874.

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]

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

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.

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

A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

Original entry on oeis.org

1, 1, 1, 1, 25, 49, 18, 25, 32, 1, 841, 1681, 49, 169, 289, 50, 169, 288, 49, 289, 529, 128, 289, 450, 98, 625, 1152, 289, 625, 961, 800, 841, 882, 162, 1681, 3200, 288, 1369, 2450, 529, 1369, 2209, 1, 28561, 57121, 49, 5329, 10609, 961, 1681, 2401, 289, 2809, 5329

Offset: 1

Jonathan Sondow, Mar 31 2017

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.
Goldbach proved that a product of 3 consecutive positive integers is never a square.
Euler proved that a product of 4 consecutive positive integers is never a square.
Erdos and Selfridge (1975) proved that a product of 2 or more consecutive positive integers is never a square or a higher power.
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=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=3 with gcd(a,d)=1.
(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

			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.

Giovanni Resta, Table of n, a(n) for n = 1..1248 (triples with product < 10^18)
P. Erdős and J.L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math., 19 (1975), 292-301.
N. Saradha, On perfect powers in products with terms from arithmetic progressions, Acta Arith., 82 (1997), 147-172.
N. Saradha, Squares in products with terms in an arithmetic progression, Acta Arith., 86 (1998), 27-43.

Cf. A046176, A078522, A284874, A284876.

N:= 10^11: # to get all triples where the product <= N
Res:= [1,0]:
for a from 1 to floor(N^(1/3)) do
  for d from 1 while a*(a+d)*(a+2*d) <= N do
     if igcd(a,d) = 1 and issqr(a*(a+d)*(a+2*d)) then
       Res:= Res, [a,d]
     fi
  od
od:
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])):
map(t -> (t[1],t[1]+t[2],t[1]+2*t[2]), Res); # Robert Israel, Apr 05 2017

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, {a, a + d, a + 2 d}]], {a, 1, nn}, {d, 0, nn}];
Sort[t, p[#1[[1]], #1[[2]], #1[[3]]] <
    p[#2[[1]], #2[[2]], #2[[3]]] &] // Flatten

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.

cp's OEIS Frontend

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

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A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A284666 List of 3-term arithmetic progressions of coprime positive integers whose product is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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