A116207 - OEIS

%I #16 Nov 28 2024 11:08:48

%S 493,607,629,757,17927,33247,93869,19467217,31223879,72757727,

%T 13454739732766891651472740499,40093333713615672956030023507,

%U 48089152118689474641229584727,66424317743191484432891678269

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

%C From _Robert Israel_, Nov 27 2024: (Start)

%C 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 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 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 72757727//72757734 = 85298138 * 85298143, where // denotes concatenation.

%p f:= proc(d) # terms where k+7 has d digits

%p     local S,x,R,k;

%p     S:= map(t -> rhs(op(t)), [msolve(x*(x+5) = 7, 10^d+1)]);

%p     R:= NULL:

%p     for x in S do

%p       k := (x*(x+5)-7)/(10^d+1);

%p       if ilog10(k+7) = d - 1 then R:= R,k fi

%p     od:

%p     op(sort([R]))

%p end proc:

%p map(f, [$1..31]); # _Robert Israel_, Nov 27 2024

%Y Cf. A116167, A116114, A116200, A116173, A116208, A116180, A116338.

%K nonn,base,look

%O 1,1

%A _Giovanni Resta_, Feb 06 2006