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A162734
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An alternating sum of all numbers from the n-th up to the (n+1)st isolated prime.
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3
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11, 30, 42, 50, 60, 73, 81, 86, 93, 105, 120, 129, 144, 160, 165, 170, 192, 217, 228, 242, 254, 260, 270, 285, 300, 312, 324, 334, 345, 356, 363, 370, 376, 381, 386, 393, 399, 405, 424, 441, 446, 453, 462, 473, 483, 489, 495, 501, 506, 525, 544, 552, 560
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OFFSET
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1,1
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COMMENTS
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11 followed by the average of each two consecutive non-twin primes. - Colin Barker, Jul 17 2014
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LINKS
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FORMULA
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EXAMPLE
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a(1) = -2+3-4+5-6+7-8+9-10+11-12+13-14+15-16+17-18+19-20+21-22+23 = 11.
a(2) = 23-24+25-26+27-28+29-30+31-32+33-34+35-36+37 = 30.
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MAPLE
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N:= 1000: # to get all terms where the larger non-twin <= N
Primes:= select(isprime, {seq(2*i-1, i=1..floor((N+1)/2))}):
NonTwins:= Primes minus (map(t->t+2, Primes) union map(t->t-2, Primes)):
11, seq((NonTwins[i]+NonTwins[i+1])/2, i=1..nops(NonTwins)-1); # Robert Israel, Jul 21 2014
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PROG
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(PARI)
non_twin_primes(pmax) = my(s=[]); forprime(p=2, pmax, if(!isprime(p-2) && !isprime(p+2), s=concat(s, p))); s
a162734(maxp) = my(ntp=non_twin_primes(maxp)); vector(#ntp-1, n, sum(k=ntp[n], ntp[n+1], -k*(-1)^k))
(Python)
from sympy import isprime, primerange
def nontwins(N):
return [p for p in primerange(1, N+1) if not (isprime(p-2) or isprime(p+2))]
def auptont(N): # all terms where the larger non-twin <= N
nt = nontwins(N)
return [sum((-1)**(j+1)*j for j in range(nt[i], nt[i+1]+1)) for i in range(len(nt)-1)]
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CROSSREFS
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KEYWORD
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nonn,less
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AUTHOR
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EXTENSIONS
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Replaced 55 by 60 and 447 by 446 - R. J. Mathar, Sep 23 2009
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STATUS
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approved
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