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A127337 Numbers that are the sum of 10 consecutive primes.

%I A127337 #35 Apr 24 2023 12:56:09
%S A127337 129,158,192,228,264,300,340,382,424,468,510,552,594,636,682,732,780,
%T A127337 824,870,912,954,1008,1060,1114,1164,1216,1266,1320,1376,1434,1494,
%U A127337 1546,1596,1650,1704,1752,1800,1854,1914,1974,2030,2084,2142,2192,2250,2310,2374
%N A127337 Numbers that are the sum of 10 consecutive primes.
%C A127337 a(n) is the absolute value of coefficient of x^9 of the polynomial Product_{j=0..9} (x - prime(n+j)) of degree 10; the roots of this polynomial are prime(n), ..., prime(n+9).
%H A127337 Zak Seidov, <a href="/A127337/b127337.txt">Table of n, a(n) for n = 1..1000</a>
%F A127337 a(n) = A127336(n)+A000040(n+9). - _R. J. Mathar_, Apr 24 2023
%p A127337 A127337 := proc(n)
%p A127337     local i ;
%p A127337     add(ithprime(n+i),i=0..9) ;
%p A127337 end proc:
%p A127337 seq(A127337(n),n=1..30) ; # _R. J. Mathar_, Apr 24 2023
%t A127337 a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a
%t A127337 Table[Plus@@Prime[Range[n, n + 9]], {n, 50}] (* _Alonso del Arte_, Feb 15 2011 *)
%t A127337 ListConvolve[ConstantArray[1, 10], Prime[Range[50]]]
%t A127337 Total/@Partition[Prime[Range[60]],10,1] (* _Harvey P. Dale_, Jan 31 2013 *)
%o A127337 (PARI) {m=46;k=10;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ _Klaus Brockhaus_, Jan 13 2007
%o A127337 (PARI) {m=46;k=10;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ _Klaus Brockhaus_, Jan 13 2007
%o A127337 (Magma) [&+[ NthPrime(n+k): k in [0..9] ]: n in [1..90] ]; // _Vincenzo Librandi_, Apr 03 2011
%o A127337 (Python)
%o A127337 from sympy import prime
%o A127337 def a(n): return sum(prime(n + i) for i in range(10))
%o A127337 print([a(n) for n in range(1, 48)]) # _Michael S. Branicky_, Dec 09 2021
%o A127337 (Python) # faster version for generating initial segment of sequence
%o A127337 from sympy import nextprime
%o A127337 def aupton(terms):
%o A127337     alst, plst = [], [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
%o A127337     for n in range(terms):
%o A127337         alst.append(sum(plst))
%o A127337         plst = plst[1:] + [nextprime(plst[-1])]
%o A127337     return alst
%o A127337 print(aupton(47)) # _Michael S. Branicky_, Dec 09 2021
%Y A127337 Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127338, A127339.
%K A127337 nonn
%O A127337 1,1
%A A127337 _Artur Jasinski_, Jan 11 2007
%E A127337 Edited by _Klaus Brockhaus_, Jan 13 2007