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A127491 Primes which are half of the absolute coefficients [x^2] of the 5th-order polynomials with prime roots as defined in A127489.

%I A127491 #10 Apr 23 2023 14:52:54
%S A127491 310733,426871,15722159,166492163,177861107,270396557,342955763,
%T A127491 406947461,1606837039,1908243773,2902193117,3386269021,5441167877,
%U A127491 6953015807,7671152921,10005413687,10979785673,14774655421,16546239937
%N A127491 Primes which are half of the absolute coefficients [x^2] of the 5th-order polynomials with prime roots as defined in A127489.
%C A127491 The polynomials are of the form (x-prime(i))*(x-prime(i+1))*..*(x-prime(i+4)). The quadratic terms have coefficients which are of the form -sum_{j<k<l} prime(j)*prime(k)*prime(l), summing over all 10 =C(5,3) combinations of products of three distinct primes in the range prime(i) to prime(i+4). If half of the absolute (sign-reversed) coefficient is prime, it is added to the sequence.
%e A127491 The first contribution is from the 11th polynomial, (x-prime(11)) *(x-prime(12)) *(x-prime(13)) *(x-prime(14)) *(x-prime(15)) = x^5 -199x^4 +15766x^3 -621466x^2 +12185065x -95041567,
%e A127491 where the coefficient of [x^2] is -621466. Its sign-reversed half is 310733, a prime.
%p A127491 isA127491 := proc(k)
%p A127491     local x,j,p ;
%p A127491     mul( x-ithprime(k+j),j=0..4) ;
%p A127491     expand(%) ;
%p A127491     abs(coeff(%,x,2)/2) ;
%p A127491     isprime(%)
%p A127491 end proc:
%p A127491 A127491k := proc(n)
%p A127491     option remember ;
%p A127491     if n = 0 then
%p A127491         0;
%p A127491     else
%p A127491         for k from procname(n-1)+1 do
%p A127491             if isA127491(k) then
%p A127491                 return k ;
%p A127491             end if;
%p A127491         end do:
%p A127491     end if;
%p A127491 end proc:
%p A127491 A127491 := proc(n)
%p A127491     option remember ;
%p A127491     local k ;
%p A127491     k := A127491k(n) ;
%p A127491     mul( x-ithprime(k+j),j=0..4) ;
%p A127491     expand(%) ;
%p A127491     abs(coeff(%,x,2)/2) ;
%p A127491 end proc:
%p A127491 seq(A127491(n),n=1..60) ; # _R. J. Mathar_, Apr 23 2023
%Y A127491 Cf. A127345 - A127351, A006094, A046301 - A046303, A127489, A127490.
%K A127491 nonn,less
%O A127491 1,1
%A A127491 _Artur Jasinski_, Jan 16 2007
%E A127491 Entries replaced to comply with the definition. - _R. J. Mathar_, Sep 26 2011