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%I #21 Sep 20 2024 17:07:51
%S 101,236,466,838,1330,1918,2862,3856,5350,7096,8622,10558,12654,15228,
%T 18090,21550,24916,27702,31500,35068,39298,45322,51240,56980,62398,
%U 66130,69958,77854,86230,96618,106888,115842,124342,133122,144090,152568,163282,174348
%N Coefficient of x^2 in the polynomial (x-p(n))*(x-p(n+1))*(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.
%H Harvey P. Dale, <a href="/A127348/b127348.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VietasFormulas.html">Vieta's Formulas</a>
%F a(n) = p(n)*p(n+1) + p(n)*p(n+2) + p(n)*p(n+3) + p(n+1)*p(n+2) + p(n+1)*p(n+3) + p(n+2)*p(n+3), where p(k) is the k-th prime (by Viete's formula relating the zeros and the coefficients of a polynomial). - _Emeric Deutsch_, Jan 20 2007
%e a(1)=101 because (x-2)*(x-3)*(x-5)*(x-7) = x^4 - 17x^3 + 101x^2 - 247x + 210.
%p a:=n->coeff(expand((x-ithprime(n))*(x-ithprime(n+1))*(x-ithprime(n+2))*(x-ithprime(n+3))),x,2): seq(a(n),n=1..45); # _Emeric Deutsch_, Jan 20 2007
%t Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 2] Prime[x + 3], {x, 1, 100}]
%t Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[Range[40]],4,1] (* _Harvey P. Dale_, Apr 15 2019 *)
%o (PARI) {m=35;k=3;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 (PARI) {m=35;k=3;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),2)),","))} \\ _Klaus Brockhaus_, Jan 21 2007
%Y Cf. A127345, A127346, A127347, A127349, A127350, A127351, A070934, A006094.
%K nonn
%O 1,1
%A _Artur Jasinski_, Jan 11 2007
%E Edited by _Emeric Deutsch_ and _Klaus Brockhaus_, Jan 20 2007