This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127345 #30 Feb 16 2025 08:33:04
%S A127345 31,71,167,311,551,791,1151,1655,2279,3119,3935,4871,5711,6791,8391,
%T A127345 9959,11639,13175,14831,16559,18383,20975,24071,27419,30191,32231,
%U A127345 33911,36071,40511,45791,51983,55199,60167,64199,69599,73911,79031,84311
%N A127345 a(n) = pq + pr + qr with p = prime(n), q = prime(n+1), and r = prime(n+2).
%C A127345 a(n) = coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(n+j)) of degree 3; the roots of this polynomial are prime(n), ..., prime(n+2); cf. Vieta's formulas.
%C A127345 Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2). [_Giorgio Balzarotti_, May 26 2011]
%H A127345 Harvey P. Dale, <a href="/A127345/b127345.txt">Table of n, a(n) for n = 1..1000</a>
%H A127345 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VietasFormulas.html">Vieta's Formulas</a>
%t A127345 Table[Prime[n]*Prime[n+1] + Prime[n]*Prime[n+2] + Prime[n+1]*Prime[n+2], {n, 100}]
%t A127345 Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[Range[40]],3,1] (* _Harvey P. Dale_, Sep 11 2017 *)
%o A127345 (PARI) {m=38;k=2;for(n=1,m,print1(sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j))),","))} \\ _Klaus Brockhaus_, Jan 21 2007
%o A127345 (PARI) {m=38;k=2;for(n=1,m,print1(polcoeff(prod(j=0,k,(x-prime(n+j))),1),","))} \\ _Klaus Brockhaus_, Jan 21 2007
%o A127345 (PARI) p=2;q=3;forprime(r=5,1e3,print1(p*q+p*r+q*r", ");p=q;q=r) \\ _Charles R Greathouse IV_, Jan 13 2012
%Y A127345 Cf. A127346, A127347, A127348, A127349, A127350, A127351, A070934, A006094.
%K A127345 nonn,easy
%O A127345 1,1
%A A127345 _Artur Jasinski_, Jan 11 2007
%E A127345 Edited by _Klaus Brockhaus_, Jan 21 2007