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%I #17 Aug 18 2022 10:34:06
%S 30,42,66,70,78,105,114,130,154,165,174,182,222,231,238,246,255,273,
%T 282,285,286,310,318,345,357,366,370,385,399,418,430,434,442,455,465,
%U 474,483,494,498,518,555,561,574,582,595,602,609,618,642,645,651,663,665
%N Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.
%C The number pq+pr+qr is prime only if p, q, and r are distinct. The primes of form pq+pr+qr are in A087054. A prime may have multiple representations as pq+pr+qr; for example, 2*3*13 and 3*5*7 both produce the prime 71.
%C As mentioned by Ufnarovski and Ahlander, if pq+pr+qr is prime, then the arithmetic derivative (A003415) of pqr is that prime. They conjecture that this sequence and A087054 are infinite.
%H T. D. Noe, <a href="/A189759/b189759.txt">Table of n, a(n) for n = 1..1000</a>
%H Victor Ufnarovski and Bo Ahlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%t pqr[nn_] := Module[{p=Prime[Range[PrimePi[nn/6]+1]],i,j,k,n,prod}, Sort[Reap[i=0; While[i++; p[[i]]p[[i+1]]p[[i+2]] <= nn, j=i; While[j++; p[[i]]p[[j]]p[[j+1]] <= nn, k=j; While[k++; prod=p[[i]]p[[j]]p[[k]]; prod <= nn, n=p[[i]]p[[j]]+p[[i]]p[[k]]+p[[j]]p[[k]]; If[PrimeQ[n], Sow[prod]]]]]][[2,1]]]]; pqr[1000]
%t Take[Union[Times@@@Select[Subsets[Prime[Range[30]],{3}],PrimeQ[ Total[ Times@@@Subsets[#,{2}]]]&]],60](* _Harvey P. Dale_, Dec 29 2011 *)
%Y Cf. A003415, A087054, A356475.
%K nonn
%O 1,1
%A _T. D. Noe_, Apr 27 2011