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%I #22 Mar 14 2023 16:24:27
%S 14,34,142,898,1762,5182,19042,79522,213442,359998,412162,685582,
%T 777922,1192462,1695202,2585662,4536898,5143822,5673922,7225342,
%U 12446782,12659362,12830722,17040382,17892898,18818242,20684302,25100098,32970562,37601422,46131262,48441598
%N Semiprimes which are the product of a twin prime pair minus one.
%C Subsequence of A103533 and A001358.
%C All the terms in this sequence, except a(1), are congruent to 1 (mod 3).
%H K. D. Bajpai, <a href="/A269663/b269663.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 2*A086870(n). - _Ray Chandler_, Apr 04 2016
%e a(1) = 14 = 2 * 7 that is semiprime. Also, 3 * 5 - 1 = 14 where {3,5} is a twin prime pair.
%e a(2) = 34 = 2 * 17 that is semiprime. Also, 5 * 7 - 1 = 34 where {5,7} is a twin prime pair.
%p A269663:= proc() local a, b, d; a:= ithprime(n); b:=a+2; d:=(a*b)-1; if isprime(b)and bigomega(d)=2 then return (d): fi; end: seq(A269663 (n), n=1..1000);
%t A269663= {}; Do[a = Prime[n]; b = a + 2; c = a*b - 1; If[PrimeQ[b] && PrimeOmega[c] == 2, AppendTo[A269663, c]], {n, 1000}]; A269663
%t Select[Times @@ # - 1 & /@ Transpose@{#, 2 + #} &@ Select[Prime@ Range@ 900, NextPrime@ # == # + 2 &], PrimeOmega@ # == 2 &] (* _Michael De Vlieger_, Apr 01 2016 *)
%t Select[Times@@@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&]-1,PrimeOmega[ #]==2&] (* _Harvey P. Dale_, Mar 14 2023 *)
%o (PARI) for(n = 1, 1000, p = prime(n); q = p + 2; c=(p*q) - 1; if(isprime(q) && bigomega(c)==2, print1(c, ", ")));
%o (Magma) IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ s: n in [1..1000] | IsPrime(n) and IsPrime(n+2) and IsP2(s) where s is (n * (n+2) - 1)];
%Y Cf. A001097, A001358, A001359, A006512, A086870, A103533, A118071.
%K nonn
%O 1,1
%A _K. D. Bajpai_, Mar 02 2016
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