A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)*prime(j)*prime(k).

247, 886, 2556, 6288, 12900, 22392, 40808, 63978, 105000, 161142, 216232, 294168, 385544, 507782, 658820, 858000, 1067502, 1251952, 1518910, 1783854, 2114748, 2618148, 3147710, 3696090, 4239528, 4626300, 5033232, 5898936, 6871200

Offset: 1

Author

Artur Jasinski, Jan 11 2007

Keywords

• nonn

Comments

a(n) = absolute value of the coefficient of x^1 of the polynomial Product_{j=0..3} (x - prime(n+j)) of degree 4; the roots of this polynomial are prime(n), ..., prime(n+3); cf. Vieta's formulas.

All terms with exception of the first one are even.

Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2)*prime(n+3). - Giorgio Balzarotti, May 26 2011

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Vieta's Formulas

Crossrefs

Cf. A046302, A127345, A127346, A127347, A127348, A127350, A127351, A070934, A006094.

Programs

Magma

[NthPrime(n)*NthPrime(n+1)*NthPrime(n+2) + NthPrime(n)*NthPrime(n+2)*NthPrime(n+3) + NthPrime(n)*NthPrime(n+1)* NthPrime(n+3) + NthPrime(n+1)*NthPrime(n+2)*NthPrime(n+3): n in [1..30]]; // Vincenzo Librandi, Feb 12 2018

Maple

P := select(isprime, [2, seq(i, i = 1 .. 1000, 2)]):
f := L) -> convert(L, `*`)*add(1/t, t = L):
seq(f(P[i..i+3]),i=1..nops(P)-3); # Robert Israel, Feb 11 2018

Mathematica

Table[Prime[n] Prime[n+1] Prime[n+2] + Prime[n] Prime[n+2] Prime[n+3] + Prime[n] Prime[n+1] Prime[n+3] + Prime[n+1] Prime[n+2] Prime[n+3], {n, 100}]

PARI

{m=29;h=3;for(n=1,m,print1(sum(i=n,n+h-2,sum(j=i+1,n+h-1,sum(k=j+1,n+h,prime(i)*prime(j)*prime(k)))),","))} \\ Klaus Brockhaus, Jan 21 2007

PARI

{m=29;k=3;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),1)),","))} \\ Klaus Brockhaus, Jan 21 2007

Formula

a(n) = A046302(n)*Sum_{i=n..n+3} 1/prime(i). - Robert Israel, Feb 11 2018

Extensions

Edited by Klaus Brockhaus, Jan 21 2007