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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. 1
1, 5, 8, 9, 22, 29, 45, 49, 60, 69, 87, 89, 90, 107, 114, 124, 125, 131, 134, 138, 145, 156, 171, 183, 188, 191, 203, 204, 207, 212, 219, 255, 261, 290, 298, 303, 329, 330, 343, 344, 349, 354, 378, 397, 398, 400, 403, 454, 456, 466, 474, 515, 549, 560, 570, 578 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

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

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

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.

MATHEMATICA

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 3] + Prime[x] 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

PROG

(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]))))

pr(p, n)=my(v=vector(n)); v[1]=p; for(i=2, #v, v[i]=nextprime(v[i-1]+1)); v

is(n, p=prime(n))=isprime(e4(pr(p, 5)))

v=List(); n=0; forprime(p=2, 1e4, if(is(n++, p), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jun 15 2015

CROSSREFS

Cf. A001043, A034961, A034963, A034964, A127333-A127343, A127345-A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301-A046303, A046324-A046327, A127489, A127491, A127492, A024449.

Sequence in context: A102785 A260348 A276934 * A006186 A230457 A180748

Adjacent sequences:  A127490 A127491 A127492 * A127494 A127495 A127496

KEYWORD

nonn

AUTHOR

Artur Jasinski, Jan 16 2007

EXTENSIONS

Definition and comment rephrased and examples added by R. J. Mathar, Oct 01 2009

STATUS

approved

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Last modified January 21 19:37 EST 2019. Contains 319350 sequences. (Running on oeis4.)