A353203 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.

67613590, 72808450, 125918320, 153469030, 190281850, 229119880, 328315900, 339204910, 360203140, 395961280, 447304000, 450075340, 692309530, 844334920, 861327610, 909001390, 1029358330, 1166831380, 1178236510, 1321005400, 1344348610, 1366379080, 2035500610, 2045710810, 2156564410

Offset

1,1

Comments

Other such terms are 18806843674476 and 18806855958880.

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

Links

Table of n, a(n) for n=1..25.

Example

If b = 6, then c = 25 and d = c-b = 19 (prime), so 6 is not in the sequence.
If b = 67613590, then c = 67613611, and d = c-b = 21 (neither prime nor square), so 67613590 is in the sequence.

Mathematica

c[n_]:=Module[{k=n+1}, While[GCD[n, k]!=1||PrimeQ[k], k++]; k];
Select[Range[10^8], CompositeQ[#]&&CompositeQ[c[#]-#]&&!IntegerQ[Sqrt[c[#]-#]]&]

Crossrefs

Cf. A002808, A113496, A342175.

Sequence in context: A089081 A161774 A203670 * A183707 A256274 A158890

Adjacent sequences: A353200 A353201 A353202 * A353204 A353205 A353206

Keyword

nonn

Author

Fausto Morales Díaz and Ivan N. Ianakiev, Apr 30 2022

Status

approved