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

493, 607, 629, 757, 17927, 33247, 93869, 19467217, 31223879, 72757727, 13454739732766891651472740499, 40093333713615672956030023507, 48089152118689474641229584727, 66424317743191484432891678269

Offset: 1

Author

Giovanni Resta, Feb 06 2006

Keywords

• nonn
• base
• look

Comments

From Robert Israel, Nov 27 2024: (Start)

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.

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)

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..85 (all terms with up to 113 digits).

Crossrefs

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

Programs

Maple

f:= proc(d) # terms where k+7 has d digits
    local S,x,R,k;
    S:= map(t -> rhs(op(t)), [msolve(x*(x+5) = 7, 10^d+1)]);
    R:= NULL:
    for x in S do
      k := (x*(x+5)-7)/(10^d+1);
      if ilog10(k+7) = d - 1 then R:= R,k fi
    od:
    op(sort([R]))
end proc:
map(f, [$1..31]); # Robert Israel, Nov 27 2024