A268697 - OEIS

%I #13 Sep 08 2022 08:46:15

%S 30,42,102,462,2130,2802,3930,5658,6198,6270,6870,7458,7590,8970,9042,

%T 9858,10302,11490,11778,13710,13722,13998,14322,17490,17790,18042,

%U 19470,20478,22278,22962,23910,25998,29670,30390,31722,32190,32370,32610,32802,32910,33330

%N Squarefree numbers n such that n^2 + 1 and n^2 - 1 are semiprime.

%C All terms are divisible by 6.

%C Subset of A014574. - _Robert Israel_, Feb 11 2016

%H Robert Israel, <a href="/A268697/b268697.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 30 = 2 * 3 * 5 which is squarefree. 30^2 + 1 = 901 = 17 * 53; 30^2 - 1 = 899 = 29 * 31; 901 and 899 are both semiprime.

%e a(2) = 42 = 2 * 3 * 7 which is squarefree. 42^2 + 1 = 1765 = 5 * 353; 30^2 - 1 = 1763 = 41 * 43; 1765 and 1763 are both semiprime.

%p with(numtheory):A268697 := proc(n) if issqrfree(n) and bigomega(n^2+1)=2 and bigomega(n^2-1)=2 then RETURN (n); fi; end: seq(A268697 (n), n=2..10000);

%t Select[Range[100000], SquareFreeQ[#] && PrimeOmega[#^2 + 1] == 2 && PrimeOmega[#^2 - 1] == 2 &]

%o (PARI) for(n=2, 1000, issquarefree(n) & bigomega(n^2 + 1)==2 & bigomega(n^2 - 1)==2 & print1(n, ", "))

%o (Magma) IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..50000] | IsSquarefree(n) and IsP2(n^2+1) and IsP2(n^2-1)];

%Y Cf. A001358, A005117, A014574, A039956, A268641.

%K nonn

%O 1,1

%A _K. D. Bajpai_, Feb 11 2016