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A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

%I A127493 #21 Jun 07 2025 23:56:04
%S A127493 1,5,8,9,22,29,45,49,60,69,87,89,90,107,114,124,125,131,134,138,145,
%T A127493 156,171,183,188,191,203,204,207,212,219,255,261,290,298,303,329,330,
%U A127493 343,344,349,354,378,397,398,400,403,454,456,466,474,515,549,560,570,578
%N A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.
%C A127493 A fifth-order polynomial with 5 roots which are the five consecutive primes from prime(k) onward is defined by Product_{j=0..4} (x-prime(k+j)). The sequence is a catalog of the cases where the coefficient of its linear term is prime.
%C A127493 Indices k such that e4(prime(k), prime(k+1), ..., prime(k+4)) is prime, where e4 is the elementary symmetric polynomial summing all products of four variables. - _Charles R Greathouse IV_, Jun 15 2015
%H A127493 Charles R Greathouse IV, <a href="/A127493/b127493.txt">Table of n, a(n) for n = 1..10000</a>
%e A127493 For k=2, the polynomial is (x-3)*(x-5)*(x-7)*(x-11)*(x-13) = x^5-39*x^4+574*x^3-3954*x^2+12673*x-15015, where 12673 is not prime, so k=2 is not in the sequence.
%e A127493 For k=5, the polynomial is x^5-83*x^4+2710*x^3-43490*x^2+342889*x-1062347, where 342889 is prime, so k=5 is in the sequence.
%p A127493 isA127493 := proc(k)
%p A127493     local x,j ;
%p A127493     mul( x-ithprime(k+j),j=0..4) ;
%p A127493     expand(%) ;
%p A127493     isprime(coeff(%,x,1)) ;
%p A127493 end proc:
%p A127493 A127493 := proc(n)
%p A127493     option remember ;
%p A127493     if n = 1 then
%p A127493         1;
%p A127493     else
%p A127493         for a from procname(n-1)+1 do
%p A127493             if isA127493(a) then
%p A127493                 return a;
%p A127493             end if;
%p A127493         end do:
%p A127493     end if;
%p A127493 end proc:
%p A127493 seq(A127493(n),n=1..60) ; # _R. J. Mathar_, Apr 23 2023
%t A127493 a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 2]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3]Prime[x + 4])], AppendTo[a, x]], {x, 1, 1000}]; a
%o A127493 (PARI) e4(v)=sum(i=1,#v-3, v[i]*sum(j=i+1,#v-2, v[j]*sum(k=j+1,#v-1, v[k]*vecsum(v[k+1..#v]))))
%o A127493 pr(p, n)=my(v=vector(n)); v[1]=p; for(i=2,#v, v[i]=nextprime(v[i-1]+1)); v
%o A127493 is(n,p=prime(n))=isprime(e4(pr(p,5)))
%o A127493 v=List(); n=0; forprime(p=2,1e4, if(is(n++,p), listput(v,n))); Vec(v) \\ _Charles R Greathouse IV_, Jun 15 2015
%Y A127493 Cf. A001043, A034961, A034963, A034964, A127333-A127343, A127345-A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301-A046303, A046324-A046327, A127489, A127491, A127492, A024449.
%K A127493 nonn
%O A127493 1,2
%A A127493 _Artur Jasinski_, Jan 16 2007
%E A127493 Definition and comment rephrased and examples added by _R. J. Mathar_, Oct 01 2009