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

1,1

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

Formula

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

Prog

(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)))), ", "))} \* or *\
{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
(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

Crossrefs

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

Sequence in context: A261378 A063356 A250608 * A051153 A166399 A129133

Adjacent sequences: A127346 A127347 A127348 * A127350 A127351 A127352

Keyword

nonn

Author

Artur Jasinski, Jan 11 2007

Extensions

Edited by Klaus Brockhaus, Jan 21 2007

Status

approved