A127345 a(n) = pq + pr + qr with p = prime(n), q = prime(n+1), and r = prime(n+2).

31, 71, 167, 311, 551, 791, 1151, 1655, 2279, 3119, 3935, 4871, 5711, 6791, 8391, 9959, 11639, 13175, 14831, 16559, 18383, 20975, 24071, 27419, 30191, 32231, 33911, 36071, 40511, 45791, 51983, 55199, 60167, 64199, 69599, 73911, 79031, 84311

Offset: 1

Author

Artur Jasinski, Jan 11 2007

Keywords

• nonn
• easy

Comments

a(n) = coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(n+j)) of degree 3; the roots of this polynomial are prime(n), ..., prime(n+2); cf. Vieta's formulas.

Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2). [Giorgio Balzarotti, May 26 2011]

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Vieta's Formulas

Crossrefs

Cf. A127346, A127347, A127348, A127349, A127350, A127351, A070934, A006094.

Programs

Mathematica

Table[Prime[n]*Prime[n+1] + Prime[n]*Prime[n+2] + Prime[n+1]*Prime[n+2], {n, 100}]
Total[Times@@@Subsets[#,{2}]]&/@Partition[Prime[Range[40]],3,1] (* Harvey P. Dale, Sep 11 2017 *)

PARI

{m=38;k=2;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=38;k=2;for(n=1,m,print1(polcoeff(prod(j=0,k,(x-prime(n+j))),1),","))} \\ Klaus Brockhaus, Jan 21 2007

PARI

p=2;q=3;forprime(r=5,1e3,print1(p*q+p*r+q*r", ");p=q;q=r) \\ Charles R Greathouse IV, Jan 13 2012

Extensions

Edited by Klaus Brockhaus, Jan 21 2007