A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

11, 29, 79, 271, 379, 461, 521, 631, 739, 881, 929, 1459, 1531, 1709, 2161, 2239, 2341, 2729, 3049, 3491, 3709, 4021, 4349, 4561, 4691, 5021, 5281, 5851, 5879, 6301, 6329, 6829, 7559, 8009, 9151, 10069, 10099, 10151, 10529, 10891, 11719, 11959, 11969, 13799, 14051, 14159

Offset: 1

Author

Zak Seidov, Jan 08 2011

Keywords

• nonn

Comments

From the first 10^6 primes, 6680 are terms of the sequence.

Also, all numbers k^2+1 are twice prime, and k^2+2 are thrice prime.

The number of terms less than 10^m beginning with m = 1: 0, 3, 11, 35, 160, 759, 4668, 30319, 204439, ..., .

The number of terms less than the (10^m)-th prime beginning with m = 1: 2, 7, 33, 165, 941, 6680, 48977, 373627, ..., .

Links

• Zak Seidov, Table of n, a(n) for n = 1..6680

Crossrefs

n^2 are squares in A070552, which is a subsequence of A056809 (m and m+1 are semiprimes) and A001358 (semiprimes).

The sequence is a subsequence of A048161.

Programs

Mathematica

fQ[n_] := PrimeQ[(n^2 + 1)/2] && PrimeQ[(n^2 + 2)/3]; Select[ Prime@ Range@ 1667, fQ] (* Robert G. Wilson v, Feb 26 2011 *)
Select[Range[15000],PrimeOmega[#^2+{0,1,2}]=={2,2,2}&] (* Harvey P. Dale, May 12 2025 *)

PARI

{n=10;for(i=1,10^4,n=nextprime(n+1);n2=n^2;if(2==bigomega(n2+1)&&2==bigomega(n2+2),print1(n,",")))}