A340231 - OEIS

%I #62 Dec 27 2022 12:02:45

%S 12,45,2021,3132,1456414565,3823938240,6991969920,120395120396,

%T 426436426437,902596902597,74780207478021,90902209090221,

%U 66713320846671332085,81142640598114264060,84822272598482227260,99002509969900250997,22443387868362244338786837,24905771529642490577152965

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

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

%C 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.

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

%C The next such term is A115439(1062)^2 - 4. - _David A. Corneth_, Jan 02 2021

%e a(1) = 12 = 4^2-4 = 2*6.

%e a(4) = 3132 = 56^2-4 = 54*58.

%t 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 *)

%o (Python)

%o def agen():

%o   m = 4

%o   while True:

%o     tstr = str(m*m-4)

%o     k = int(tstr[:len(tstr)//2])

%o     if tstr == str(k) + str(k+1):

%o       yield(int(tstr))

%o     m += 1

%o for an in agen(): print(an, end=", ") # _Michael S. Branicky_, Jan 02 2021

%Y Intersection of A001704 and A028347.

%Y Cf. A115439, A143206.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jan 01 2021

%E a(13)-a(16) from _Michael S. Branicky_, Jan 02 2021

%E a(17)-a(18) from _David A. Corneth_, Jan 02 2021