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

310733, 426871, 15722159, 166492163, 177861107, 270396557, 342955763, 406947461, 1606837039, 1908243773, 2902193117, 3386269021, 5441167877, 6953015807, 7671152921, 10005413687, 10979785673, 14774655421, 16546239937

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..19.

Example

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

Maple

isA127491 := proc(k)
    local x, j, p ;
    mul( x-ithprime(k+j), j=0..4) ;
    expand(%) ;
    abs(coeff(%, x, 2)/2) ;
    isprime(%)
end proc:
A127491k := proc(n)
    option remember ;
    if n = 0 then
        0;
    else
        for k from procname(n-1)+1 do
            if isA127491(k) then
                return k ;
            end if;
        end do:
    end if;
end proc:
A127491 := proc(n)
    option remember ;
    local k ;
    k := A127491k(n) ;
    mul( x-ithprime(k+j), j=0..4) ;
    expand(%) ;
    abs(coeff(%, x, 2)/2) ;
end proc:
seq(A127491(n), n=1..60) ; # R. J. Mathar, Apr 23 2023

Crossrefs

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

Sequence in context: A206276 A377953 A251304 * A205204 A205062 A257015

Adjacent sequences: A127488 A127489 A127490 * A127492 A127493 A127494

Keyword

nonn,less

Author

Artur Jasinski, Jan 16 2007

Extensions

Entries replaced to comply with the definition. - R. J. Mathar, Sep 26 2011

Status

approved