|
%I #20 Mar 01 2024 17:12:30
%S 35,65,77,133,143,161,209,215,221,235,265,335,377,391,403,413,451,517,
%T 527,551,553,565,583,623,635,667,685,707,721,731,763,779,793,817,835,
%U 851,871,893,917,923,965,1007,1057,1067,1133,1147,1157,1207,1243,1247,1271,1273,1313,1333,1337,1363,1385
%N Squarefree semiprimes p*q such that (p*q+1+p-q)/2 and (p*q+1+q-p)/2 are prime.
%C All terms are odd.
%H Robert Israel, <a href="/A354502/b354502.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 77 = 7*11 is a term because p = 7, q = 11, (7*11+1+11-7)/2 = 41 and (7*11+1+7-11)/2 = 37 are prime.
%p filter:= proc(n) local p,q,t,k;
%p if issqr(n) or numtheory:-bigomega(n) <> 2 then return false fi;
%p p,q:= op(numtheory:-factorset(n));
%p isprime((n+1+p-q)/2) and isprime((n+1+q-p)/2)
%p end proc:
%p select(filter, [seq(i,i=3..10000,2)]);
%t Select[Range[1, 1400, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1} && PrimeQ[((p = f[[1, 1]])*(q = f[[2, 1]]) + 1 + p - q)/2] && PrimeQ[(p*q + 1 + q - p)/2] &] (* _Amiram Eldar_, Aug 16 2022 *)
%t sfspQ[n_]:=Module[{p,q},{p,q}=FactorInteger[n][[;;,1]];AllTrue[{(n+1+p-q)/2,(n+1+q-p)/2},PrimeQ]]; Select[Range[1500],SquareFreeQ[#]&&PrimeOmega[#]==2&&sfspQ[#]&] (* _Harvey P. Dale_, Mar 01 2024 *)
%Y Cf. A006881.
%K nonn,less
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Aug 15 2022
|
