A115556 Numbers whose square is the concatenation of two numbers 9*m and m.

12857142857142857142857142857142857143, 25714285714285714285714285714285714286, 117391304347826086956521739130434782608695652173913043478261

Offset

1,1

Comments

a(4)=156521739130434782608695652173913043478260869565217391304348.

From Robert Israel, Aug 24 2023: (Start)

If 9 * 10^d + 1 = a^2 * b with a > 1, then a * b * c is a term if a^2/(90 + 10^(1-d)) < c^2 < a^2/(9 + 10^(-d)). For example, 9 * 10^d + 1 is divisible by 7^2 for d == 37 (mod 42), and then (9 * 10^d + 1)/7 and 2*(9 * 10^d + 1)/7 are terms. In particular, the sequence is infinite. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..12

Maple

F:= proc(d) local R, F, t, b, r, q, s, m0, x0, k;
     R:= NULL;
     F:= ifactors(9*10^d+1)[2];
     b:= mul(t[1]^floor(t[2]/2), t=F);
     for r in numtheory:-divisors(b) do
       x0:= (9*10^d+1)/r;
       m0:= x0/r;
       for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
         R:= R, x0*k;
       od
     od;
       R
end proc:
sort(map(F, [$1..90])); # Robert Israel, Aug 24 2023

Crossrefs

Cf. A009474, A102567, A106497, A115527 - A115555.

Sequence in context: A217419 A246289 A095468 * A173907 A095470 A329659

Adjacent sequences: A115553 A115554 A115555 * A115557 A115558 A115559

Keyword

nonn,base,bref

Author

Giovanni Resta, Jan 25 2006

Extensions

Definition modified by Georg Fischer, Jul 26 2019

Status

approved