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

1,1

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)

If k+7 has d digits, the concatenation is k//k+7 = 10^d*k+k+7 = m*(m+5), therefore k//k+1 = 10^d*k+k+1 = (m-1)*(m+6), so there is a large overlap with A116167. Also k//k-7 = 10^d*k+k-7= (m-2)*(m+7), so there is a large overlap with A116114. - R. J. Mathar, May 22 2026

Links

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

Example

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

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

Crossrefs

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

Sequence in context: A198531 A300058 A251540 * A363391 A264327 A045007

Adjacent sequences: A116204 A116205 A116206 * A116208 A116209 A116210

Keyword

nonn,base,look,changed

Author

Giovanni Resta, Feb 06 2006

Status

approved