A115432 Numbers k such that the concatenation of k with k-4 gives a square.

65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604

Offset

1,1

Comments

The terms of this sequence (k//k-4 = m*m), A116104 (k//k-8 = m*(m+4)) and A116121 (k//k-5 = m*(m+2)) agree as long as the two concatenated numbers k and k-x have the same length. This condition is satisfied for the given terms of all three sequences. - Georg Fischer, Sep 12 2022

From Robert Israel, Sep 13 2023: (Start)

Numbers k of the form (y^2+4)/(10^d + 1) where 10^(d-1) <= k - 4 < 10^d and y is a square root of -4 mod (10^d + 1).

Includes 10^(2*d) - 4*10^d + 5 for all d >= 1, as the concatenation of this with 10^(2*d) - 4*10^d + 1 is 10^(4*d) - 4 * 10^(3*d) + 6 * 10^(2*d) - 4 * 10^d + 1 = (10^d - 1)^4.

This is the same sequence as A116104 and A116121. The only possible differences would be if 10^(d-1) + 4 <= k <= 10^(d-1) + 7 or 10^d + 4 <= k <= 10^d + 7, so that k - 4 and k - 8 have different numbers of digits.

But in none of those cases can (10^d + 1)*k - 4 be a square:

If k = 10^(d-1) + 4 or 10^d + 4, (10^d + 1)*k - 4 == 6 (mod 9).

If k = 10^(d-1) + 5 or 10^d + 5, (10^d + 1)*k - 4 == 2 (mod 3).

If k = 10^(d-1) + 6 or 10^d + 6, (10^d + 1)*k - 4 == 2 (mod 10).

If k = 10^(d-1) + 7 or 10^d + 7, (10^d + 1)*k - 4 == 3 (mod 10). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..2805

Example

9605_9601 = 9801^2.

Maple

f:= proc(d) uses NumberTheory; local m, r;
  m:= 10^d + 1;
  if QuadraticResidue(-4, m) = -1 then return NULL fi;
  r:= ModularSquareRoot(-4, m);
  op(sort(select(t -> t >= 10^(d-1)+4 and t < 10^d+4, map(t -> ((r*t mod m)^2+4)/m, convert(RootsOfUnity(2, m), list)))))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 12 2023

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.

Almost the same as A116104 and A116121.

Sequence in context: A146756 A202386 A294955 * A116104 A116121 A118707

Adjacent sequences: A115429 A115430 A115431 * A115433 A115434 A115435

Keyword

base,nonn

Author

Giovanni Resta, Jan 25 2006

Status

approved