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A340465
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Primes of the form prime(i)*prime(i+1)+prime(i+2)*prime(i+3)+...+prime(k-1)*prime(k).
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2
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41, 313, 2137, 6569, 7853, 10133, 10847, 12401, 13757, 14747, 17569, 17911, 24001, 24049, 27901, 31307, 38729, 43177, 43961, 44819, 51607, 69191, 81517, 88379, 104683, 107099, 130631, 137177, 138239, 145967, 154487, 154723, 158777, 162947, 175463, 184409, 192853, 196169, 232499, 243137, 261983
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history;
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OFFSET
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1,1
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COMMENTS
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A prime that has more than one expression of the given form is included only once. The first such prime is a(14353) = 6858604873 = 1979*1987+...+7109*7121 = 19949*19961+...+20231*20233.
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..10000
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EXAMPLE
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a(1) = 2*3+5*7 = 41.
a(2) = 3*5+7*11+13*17 = 313.
a(3) = 17*19+23*29+31*37 = 2137.
a(4) = 5*7+11*13+17*19+23*29+31*37+41*43+47*53 = 6569.
a(5) = 41*43+47*53+59*61 = 7853.
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MAPLE
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S1:= [0, seq(ithprime(2*i)*ithprime(2*i+1), i=1..100)]:
P1:= ListTools:-PartialSums(S1):
S2:= [0, seq(ithprime(2*i-1)*ithprime(2*i), i=1..100)]:
P2:= ListTools:-PartialSums(S2):
M:= 2*max(S1):
S:= select(t -> t < M and isprime(t), {seq(seq(P1[i]-P1[j], j=i mod 2 + 1 .. i-2, 2), i=1..101)} union {seq(seq(P2[i]-P2[j], j=i mod 2 + 1..i-2, 2), i=1..101)} union {seq(P2[i], i=1..101, 2)}):
sort(convert(S, list));
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PROG
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(Python)
from sympy import isprime, nextprime, prime
def sp2(lst):
ans = 0
for i in range(0, len(lst), 2): ans += lst[i]*lst[i+1]
return ans
def aupto(nn):
alst, i = [], 1
while True:
consec2i = [prime(j+1) for j in range(2*i)]; sp = sp2(consec2i)
if sp > nn: break
while sp <= nn:
if isprime(sp): alst.append(sp)
consec2i = consec2i[1:] + [nextprime(consec2i[-1])]; sp = sp2(consec2i)
i += 1
return sorted(alst)
print(aupto(261983)) # Michael S. Branicky, Jan 08 2021
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CROSSREFS
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Includes A340464.
Sequence in context: A175110 A096170 A277201 * A282867 A222990 A300775
Adjacent sequences: A340462 A340463 A340464 * A340466 A340467 A340468
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KEYWORD
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nonn
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AUTHOR
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J. M. Bergot and Robert Israel, Jan 08 2021
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STATUS
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approved
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