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Numbers that are the sum of 10 consecutive primes.
15

%I #35 Apr 24 2023 12:56:09

%S 129,158,192,228,264,300,340,382,424,468,510,552,594,636,682,732,780,

%T 824,870,912,954,1008,1060,1114,1164,1216,1266,1320,1376,1434,1494,

%U 1546,1596,1650,1704,1752,1800,1854,1914,1974,2030,2084,2142,2192,2250,2310,2374

%N Numbers that are the sum of 10 consecutive primes.

%C 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 Zak Seidov, <a href="/A127337/b127337.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A127336(n)+A000040(n+9). - _R. J. Mathar_, Apr 24 2023

%p A127337 := proc(n)

%p local i ;

%p add(ithprime(n+i),i=0..9) ;

%p end proc:

%p seq(A127337(n),n=1..30) ; # _R. J. Mathar_, Apr 24 2023

%t a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a

%t Table[Plus@@Prime[Range[n, n + 9]], {n, 50}] (* _Alonso del Arte_, Feb 15 2011 *)

%t ListConvolve[ConstantArray[1, 10], Prime[Range[50]]]

%t Total/@Partition[Prime[Range[60]],10,1] (* _Harvey P. Dale_, Jan 31 2013 *)

%o (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 (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 (Magma) [&+[ NthPrime(n+k): k in [0..9] ]: n in [1..90] ]; // _Vincenzo Librandi_, Apr 03 2011

%o (Python)

%o from sympy import prime

%o def a(n): return sum(prime(n + i) for i in range(10))

%o print([a(n) for n in range(1, 48)]) # _Michael S. Branicky_, Dec 09 2021

%o (Python) # faster version for generating initial segment of sequence

%o from sympy import nextprime

%o def aupton(terms):

%o alst, plst = [], [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

%o for n in range(terms):

%o alst.append(sum(plst))

%o plst = plst[1:] + [nextprime(plst[-1])]

%o return alst

%o print(aupton(47)) # _Michael S. Branicky_, Dec 09 2021

%Y Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127338, A127339.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jan 11 2007

%E Edited by _Klaus Brockhaus_, Jan 13 2007