A116126 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 8.
10, 6752089, 6448802889351008245, 18894512461523256139943105859903480218905, 31958875438439894736354375209245786214798
Offset: 1
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
F:= proc(d) local R, t,alpha, beta, gamma, delta, B,C,n,m,i0,i,gamma0, delta0;
R:= NULL;
t:= 10^d+1;
for alpha in numtheory:-divisors(t) do
beta:= t/alpha;
if igcd(alpha,beta) > 1 then next fi;
delta0:= 6/beta mod alpha;
gamma0:= (beta*delta0-6)/alpha;
B:= 2*alpha*gamma0 + 6;
C:= gamma0*delta0 - 10^(d-1) - 7;
if C < 0 then i0:= 0 else i0:= ceil((-B + sqrt(B^2-4*t*C))/(2*t)) fi;
for i from i0 do
gamma:= gamma0 + i*beta;
delta:= delta0 + i*alpha;
m:= gamma*delta;
if m -7 >= 10^d then break fi;
if m - 7 >= 10^(d-1) then R:= R, alpha*gamma-1 fi;
od
od;
sort(convert({R},list))
end proc:
seq(op(F(d)),d=1..10); # Robert Israel, Aug 22 2023
a[n_] := Module[{solutions = {}, kvalues, e = 2}, While[Length[solutions] < n, sol = Solve[{a*b == 10^e + 1, 10^(e - 1) <= c*d < 10^e, a*c + 6 == b*d, a > 0, b > 0, c > 0, d > 0}, {a, b, c, d}, Integers]; kvalues = (a*c - 1) /. sol; solutions = Union[solutions, kvalues]; e++]; Take[solutions, n]]; a[26] (* Robert P. P. McKone, Aug 22 2023 *)
78317866 * 78317874 = 61336887//61336884, where // denotes concatenation.
8217108 * 8217112 = 6752089//6752096, where // denotes concatenation.
8809 * 8818 = 7767//7762, where // denotes concatenation.
Comments