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

%I A340231 #62 Dec 27 2022 12:02:45
%S A340231 12,45,2021,3132,1456414565,3823938240,6991969920,120395120396,
%T A340231 426436426437,902596902597,74780207478021,90902209090221,
%U A340231 66713320846671332085,81142640598114264060,84822272598482227260,99002509969900250997,22443387868362244338786837,24905771529642490577152965
%N A340231 Numbers of the form m^2-4 and also equal to some k concatenated with k+1.
%C A340231 All the terms have an even number of digits, but there is no term with 6, 8, 16, 18, 22, 24, ... digits.
%C A340231 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 A340231 a(3) = 2021 = 43*47 is A143206(6), the product of a cousin prime pair.
%C A340231 The next such term is A115439(1062)^2 - 4. - _David A. Corneth_, Jan 02 2021
%e A340231 a(1) = 12 = 4^2-4 = 2*6.
%e A340231 a(4) = 3132 = 56^2-4 = 54*58.
%t A340231 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 A340231 (Python)
%o A340231 def agen():
%o A340231   m = 4
%o A340231   while True:
%o A340231     tstr = str(m*m-4)
%o A340231     k = int(tstr[:len(tstr)//2])
%o A340231     if tstr == str(k) + str(k+1):
%o A340231       yield(int(tstr))
%o A340231     m += 1
%o A340231 for an in agen(): print(an, end=", ") # _Michael S. Branicky_, Jan 02 2021
%Y A340231 Intersection of A001704 and A028347.
%Y A340231 Cf. A115439, A143206.
%K A340231 nonn,base
%O A340231 1,1
%A A340231 _Bernard Schott_, Jan 01 2021
%E A340231 a(13)-a(16) from _Michael S. Branicky_, Jan 02 2021
%E A340231 a(17)-a(18) from _David A. Corneth_, Jan 02 2021