login
Let b be a composite number, c be the smallest composite number greater than b and coprime to b, and d = c-b. This sequence contains all b such that d is neither a prime nor a square.
1

%I #19 Jun 22 2022 20:23:32

%S 67613590,72808450,125918320,153469030,190281850,229119880,328315900,

%T 339204910,360203140,395961280,447304000,450075340,692309530,

%U 844334920,861327610,909001390,1029358330,1166831380,1178236510,1321005400,1344348610,1366379080,2035500610,2045710810,2156564410

%N Let b be a composite number, c be the smallest composite number greater than b and coprime to b, and d = c-b. This sequence contains all b such that d is neither a prime nor a square.

%C Other such terms are 18806843674476 and 18806855958880.

%C a(n) is even. Proof: If a(n) = b is odd then c = a(n) + 1 where gcd(b, c) = 1 and d = c-b = 1 which is a square. Contradiction. - _David A. Corneth_, May 01 2022

%e If b = 6, then c = 25 and d = c-b = 19 (prime), so 6 is not in the sequence.

%e If b = 67613590, then c = 67613611, and d = c-b = 21 (neither prime nor square), so 67613590 is in the sequence.

%t c[n_]:=Module[{k=n+1},While[GCD[n,k]!=1||PrimeQ[k],k++];k];

%t Select[Range[10^8],CompositeQ[#]&&CompositeQ[c[#]-#]&&!IntegerQ[Sqrt[c[#]-#]]&]

%Y Cf. A002808, A113496, A342175.

%K nonn

%O 1,1

%A Fausto Morales Díaz and _Ivan N. Ianakiev_, Apr 30 2022