cp's OEIS Frontend

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

%I A115556 #22 Aug 30 2023 20:52:06
%S A115556 12857142857142857142857142857142857143,
%T A115556 25714285714285714285714285714285714286,
%U A115556 117391304347826086956521739130434782608695652173913043478261
%N A115556 Numbers whose square is the concatenation of two numbers 9*m and m.
%C A115556 a(4)=156521739130434782608695652173913043478260869565217391304348.
%C A115556 From _Robert Israel_, Aug 24 2023: (Start)
%C A115556 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)
%H A115556 Robert Israel, <a href="/A115556/b115556.txt">Table of n, a(n) for n = 1..12</a>
%p A115556 F:= proc(d) local R,F,t,b,r,q,s,m0,x0,k;
%p A115556      R:= NULL;
%p A115556      F:= ifactors(9*10^d+1)[2];
%p A115556      b:= mul(t[1]^floor(t[2]/2),t=F);
%p A115556      for r in numtheory:-divisors(b) do
%p A115556        x0:= (9*10^d+1)/r;
%p A115556        m0:= x0/r;
%p A115556        for k from ceil(sqrt(10^(d-1)/m0)) to floor(sqrt(10^d/m0)) do
%p A115556          R:= R, x0*k;
%p A115556        od
%p A115556      od;
%p A115556        R
%p A115556 end proc:
%p A115556 sort(map(F, [$1..90])); # _Robert Israel_, Aug 24 2023
%Y A115556 Cf. A009474, A102567, A106497, A115527 - A115555.
%K A115556 nonn,base,bref
%O A115556 1,1
%A A115556 _Giovanni Resta_, Jan 25 2006
%E A115556 Definition modified by _Georg Fischer_, Jul 26 2019