A264805 Numbers n not divisible by 100 such that both concat(1,n) and concat(6,n) are squares.

80625, 890625, 724297265625, 9774697265625, 6447206984619140625, 56747553669171295166015625, 492518329910900929355621337890625, 4198069146906637888238489627838134765625, 34930039085994014667329000271856784820556640625, 280959098499075220855234768279851414263248443603515625

Offset

1,1

Comments

All terms == 75 mod 90. It appears that all terms == 8625 mod 9000.

All members of the sequence are obtained as follows.

Let f(t) = 25 t - 7/2 + 1/(16 t), and 0 < t_1 < t_2 < t_3 < t_4 where f(t_1) = f(t_4) = 1 and f(t_2) = f(t_3) = 1/10.

If -log(t_2)/log(25) -1 <= frac(d*log(10)/log(25)) < -log(t_1)/log(25) - 1, then

f(t)*10^d is in the sequence, where a = floor(d*log(10)/log(25)) - 1 and t = 25^a/10^d.

If -log(t_4)/log(25) < frac(d*log(10)/log(25)) <= -log(t_3)/log(25), then

f(t)*10^d is in the sequence, where a = floor(d*log(10)/log(25)) and t = 25^a/10^d.

By the equidistribution theorem and irrationality of log(10)/log(25), there are infinitely many terms of each type.

Links

Robert Israel, Table of n, a(n) for n = 1..177

Example

For n=1, 180625 = 425^2 and 680625 = 825^2 are squares, so 80625 is in the sequence.

Maple

f:= t -> 25*t - 7/2 + 1/(16*t):
t1, t2, t3, t4:= 9/100-(1/50)*14^(1/2), 9/125-(1/500)*671^(1/2), 9/125+(1/500)*671^(1/2), 9/100+(1/50)*14^(1/2):
v1, v2, v3, v4:= -log(t1)/log(25)-1, -log(t2)/log(25)-1, -log(t3)/log(25), -log(t4)/log(25):
A:= NULL:
for d from 1 to 100 do
  v:= frac(d*log(10)/log(25));
  if is(v >= v2 and v < v1) then
    a:= d*log(10)/log(25) - v - 1;
    A:= A, 10^d*f(25^a/10^d);
  elif is(v > v4 and v <= v3) then
    a:= d*log(10)/log(25) - v;
    A:= A, 10^d*f(25^a/10^d);
  fi
od:
A;

Prog

(PARI) is_A264805(n)=issquare(10^#digits(n)+n)&&issquare(6*10^#digits(n)+n) \\ M. F. Hasler, Nov 25 2015

Crossrefs

Sequence in context: A347906 A329188 A095946 * A050517 A069304 A157664

Adjacent sequences: A264802 A264803 A264804 * A264806 A264807 A264808

Keyword

nonn,base

Author

Robert Israel, Nov 25 2015

Status

approved