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Numbers that are the sum of 12 consecutive primes.
9

%I #22 Sep 08 2022 08:45:29

%S 197,236,276,318,364,412,460,510,562,612,662,714,766,822,880,936,990,

%T 1040,1092,1152,1212,1276,1336,1402,1464,1524,1586,1650,1716,1786,

%U 1854,1918,1980,2040,2100,2162,2234,2304,2370,2436,2502,2564,2634,2700,2770

%N Numbers that are the sum of 12 consecutive primes.

%C a(n) = absolute value of coefficient of x^11 of the polynomial Product_{j=0..11} (x - prime(n+j)) of degree 12; the roots of this polynomial are prime(n), ..., prime(n+11).

%t a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 11}]], {x, 1, 50}]; a

%t Total/@Partition[Prime[Range[60]],12,1] (* _Harvey P. Dale_, May 05 2018 *)

%o (PARI) {m=45;k=12;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ _Klaus Brockhaus_, Jan 13 2007

%o (PARI) {m=45;k=12;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ _Klaus Brockhaus_, Jan 13 2007

%o (Magma) [&+[ NthPrime(n+k): k in [0..11] ]: n in [1..100] ]; // _Vincenzo Librandi_, Apr 03 2011

%Y Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jan 11 2007

%E Edited by _Klaus Brockhaus_, Jan 13 2007