A340231 Numbers of the form m^2-4 and also equal to some k concatenated with k+1.

12, 45, 2021, 3132, 1456414565, 3823938240, 6991969920, 120395120396, 426436426437, 902596902597, 74780207478021, 90902209090221, 66713320846671332085, 81142640598114264060, 84822272598482227260, 99002509969900250997, 22443387868362244338786837, 24905771529642490577152965

Offset: 1

Author

Bernard Schott, Jan 01 2021

Keywords

• nonn
• base

Comments

All the terms have an even number of digits, but there is no term with 6, 8, 16, 18, 22, 24, ... digits.

The values of m are A115439, because a(n) = m^2-4 and a(n) = k|k+1 <==> a(n)+4 = m^2 and a(n)+4 = k|k+5 <==> m^2 = k|k+5, where | denotes concatenation.

a(3) = 2021 = 43*47 is A143206(6), the product of a cousin prime pair.

The next such term is A115439(1062)^2 - 4. - David A. Corneth, Jan 02 2021

Examples

			a(1) = 12 = 4^2-4 = 2*6.
a(4) = 3132 = 56^2-4 = 54*58.
		

Crossrefs

Intersection of A001704 and A028347.

Cf. A115439, A143206.

Programs

Mathematica

Select[Table[n 10^IntegerLength[n]+n+1,{n,10^6}],IntegerQ[Sqrt[#+4]]&] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Dec 27 2022 *)

Python

def agen():
  m = 4
  while True:
    tstr = str(m*m-4)
    k = int(tstr[:len(tstr)//2])
    if tstr == str(k) + str(k+1):
      yield(int(tstr))
    m += 1
for an in agen(): print(an, end=", ") # Michael S. Branicky, Jan 02 2021

Extensions

a(13)-a(16) from Michael S. Branicky, Jan 02 2021

a(17)-a(18) from David A. Corneth, Jan 02 2021