A127491 - OEIS

%I #10 Apr 23 2023 14:52:54

%S 310733,426871,15722159,166492163,177861107,270396557,342955763,

%T 406947461,1606837039,1908243773,2902193117,3386269021,5441167877,

%U 6953015807,7671152921,10005413687,10979785673,14774655421,16546239937

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

%C 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 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 where the coefficient of [x^2] is -621466. Its sign-reversed half is 310733, a prime.

%p isA127491 := proc(k)

%p     local x,j,p ;

%p     mul( x-ithprime(k+j),j=0..4) ;

%p     expand(%) ;

%p     abs(coeff(%,x,2)/2) ;

%p     isprime(%)

%p end proc:

%p A127491k := proc(n)

%p     option remember ;

%p     if n = 0 then

%p         0;

%p     else

%p         for k from procname(n-1)+1 do

%p             if isA127491(k) then

%p                 return k ;

%p             end if;

%p         end do:

%p     end if;

%p end proc:

%p A127491 := proc(n)

%p     option remember ;

%p     local k ;

%p     k := A127491k(n) ;

%p     mul( x-ithprime(k+j),j=0..4) ;

%p     expand(%) ;

%p     abs(coeff(%,x,2)/2) ;

%p end proc:

%p seq(A127491(n),n=1..60) ; # _R. J. Mathar_, Apr 23 2023

%Y Cf. A127345 - A127351, A006094, A046301 - A046303, A127489, A127490.

%K nonn,less

%O 1,1

%A _Artur Jasinski_, Jan 16 2007

%E Entries replaced to comply with the definition. - _R. J. Mathar_, Sep 26 2011