A368670 Numbers k such that k, k + 1, k + 2, and k + 4 are all semiprimes.

141, 201, 213, 217, 301, 1137, 1345, 1401, 1761, 1837, 1893, 1941, 1981, 2101, 3097, 3865, 3957, 4413, 4533, 4881, 5997, 6157, 6241, 7113, 7141, 7165, 7401, 7977, 8185, 8257, 8913, 9753, 9985, 10117, 11013, 11181, 11377, 11757, 12057, 13953, 14037, 14253, 14917, 14977, 14997, 16177, 16293, 16437, 16593

Offset: 1

Author

Robert Israel, Jan 02 2024

Keywords

• nonn

Comments

k, k + 1, k + 2 and k + 3 can't all be semiprimes, as one of them is divisible by 4.

All terms == 1 (mod 4).

Examples

			a(3) = 213 is a term because 213 = 3 * 71, 214 = 2 * 107, 215 = 5 * 43 and 217 = 7 * 31 are all semiprimes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A056809, A070552.

Programs

Maple

select(t -> numtheory:-bigomega(t)=2 and numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t+2)=2 and numtheory:-bigomega(t+4)=2, 4 * [$1..10000] +~ 1);

Mathematica

Select[Range[17000], PrimeOmega[#] == PrimeOmega[#+1] == PrimeOmega[#+2] == PrimeOmega[#+4] == 2 &] (* Stefano Spezia, Jan 02 2024 *)