A127348 Coefficient of x^2 in the polynomial (x-p(n))*(x-p(n+1))*(x-p(n+2))*(x-p(n+3)), where p(k) is the k-th prime.

101, 236, 466, 838, 1330, 1918, 2862, 3856, 5350, 7096, 8622, 10558, 12654, 15228, 18090, 21550, 24916, 27702, 31500, 35068, 39298, 45322, 51240, 56980, 62398, 66130, 69958, 77854, 86230, 96618, 106888, 115842, 124342, 133122, 144090, 152568, 163282, 174348

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Vieta's Formulas

Formula

a(n) = p(n)*p(n+1) + p(n)*p(n+2) + p(n)*p(n+3) + p(n+1)*p(n+2) + p(n+1)*p(n+3) + p(n+2)*p(n+3), where p(k) is the k-th prime (by Viete's formula relating the zeros and the coefficients of a polynomial). - Emeric Deutsch, Jan 20 2007

Example

a(1)=101 because (x-2)*(x-3)*(x-5)*(x-7) = x^4 - 17x^3 + 101x^2 - 247x + 210.

Maple

a:=n->coeff(expand((x-ithprime(n))*(x-ithprime(n+1))*(x-ithprime(n+2))*(x-ithprime(n+3))), x, 2): seq(a(n), n=1..45); # Emeric Deutsch, Jan 20 2007

Mathematica

Table[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 2] Prime[x + 3], {x, 1, 100}]
Total[Times@@@Subsets[#, {2}]]&/@Partition[Prime[Range[40]], 4, 1] (* Harvey P. Dale, Apr 15 2019 *)

Prog

(PARI) {m=35; k=3; for(n=1, m, print1(sum(i=n, n+k-1, sum(j=i+1, n+k, prime(i)*prime(j))), ", "))} \\ Klaus Brockhaus, Jan 21 2007
(PARI) {m=35; k=3; for(n=1, m, print1(abs(polcoeff(prod(j=0, k, (x-prime(n+j))), 2)), ", "))} \\ Klaus Brockhaus, Jan 21 2007

Crossrefs

Cf. A127345, A127346, A127347, A127349, A127350, A127351, A070934, A006094.

Sequence in context: A142910 A259739 A165379 * A061191 A085610 A142530

Adjacent sequences: A127345 A127346 A127347 * A127349 A127350 A127351

Keyword

nonn

Author

Artur Jasinski, Jan 11 2007

Extensions

Edited by Emeric Deutsch and Klaus Brockhaus, Jan 20 2007

Status

approved