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A116207 Numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 5.

%I A116207 #16 Nov 28 2024 11:08:48
%S A116207 493,607,629,757,17927,33247,93869,19467217,31223879,72757727,
%T A116207 13454739732766891651472740499,40093333713615672956030023507,
%U A116207 48089152118689474641229584727,66424317743191484432891678269
%N A116207 Numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 5.
%C A116207 From _Robert Israel_, Nov 27 2024: (Start)
%C A116207 If 10^d + 1 has a prime factor p such that 53 is not a square mod p, then there are no terms k where k + 7 has d digits.
%C A116207 For example, there are no terms where d == 2 (mod 4), since in that case 10^d + 1 is divisible by 101, and 53 is not a square mod 101. (End)
%H A116207 Robert Israel, <a href="/A116207/b116207.txt">Table of n, a(n) for n = 1..85</a> (all terms with up to 113 digits).
%e A116207 72757727//72757734 = 85298138 * 85298143, where // denotes concatenation.
%p A116207 f:= proc(d) # terms where k+7 has d digits
%p A116207     local S,x,R,k;
%p A116207     S:= map(t -> rhs(op(t)), [msolve(x*(x+5) = 7, 10^d+1)]);
%p A116207     R:= NULL:
%p A116207     for x in S do
%p A116207       k := (x*(x+5)-7)/(10^d+1);
%p A116207       if ilog10(k+7) = d - 1 then R:= R,k fi
%p A116207     od:
%p A116207     op(sort([R]))
%p A116207 end proc:
%p A116207 map(f, [$1..31]); # _Robert Israel_, Nov 27 2024
%Y A116207 Cf. A116167, A116114, A116200, A116173, A116208, A116180, A116338.
%K A116207 nonn,base,look
%O A116207 1,1
%A A116207 _Giovanni Resta_, Feb 06 2006