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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).

%I A127349 #32 Feb 16 2025 08:33:04
%S A127349 247,886,2556,6288,12900,22392,40808,63978,105000,161142,216232,
%T A127349 294168,385544,507782,658820,858000,1067502,1251952,1518910,1783854,
%U A127349 2114748,2618148,3147710,3696090,4239528,4626300,5033232,5898936,6871200
%N 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).
%C A127349 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.
%C A127349 All terms with exception of the first one are even.
%C A127349 Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2)*prime(n+3). - _Giorgio Balzarotti_, May 26 2011
%H A127349 Robert Israel, <a href="/A127349/b127349.txt">Table of n, a(n) for n = 1..10000</a>
%H A127349 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VietasFormulas.html">Vieta's Formulas</a>
%F A127349 a(n) = A046302(n)*Sum_{i=n..n+3} 1/prime(i). - _Robert Israel_, Feb 11 2018
%p A127349 P := select(isprime, [2, seq(i, i = 1 .. 1000, 2)]):
%p A127349 f := L) -> convert(L, `*`)*add(1/t, t = L):
%p A127349 seq(f(P[i..i+3]),i=1..nops(P)-3); # _Robert Israel_, Feb 11 2018
%t A127349 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}]
%o A127349 (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
%o A127349 (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
%o A127349 (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
%Y A127349 Cf. A046302, A127345, A127346, A127347, A127348, A127350, A127351, A070934, A006094.
%K A127349 nonn
%O A127349 1,1
%A A127349 _Artur Jasinski_, Jan 11 2007
%E A127349 Edited by _Klaus Brockhaus_, Jan 21 2007