A162734 An alternating sum of all numbers from the n-th up to the (n+1)st isolated prime.

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

Offset

1,1

Comments

11 followed by the average of each two consecutive non-twin primes. - Colin Barker, Jul 17 2014

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = sum_{j= A007510(n).. A007510(n+1)} (-1)^(j+1)*j = A001057(A007510(n+1))-A001057(A007510(n)-1).

Example

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.

Maple

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

Prog

(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))
a162734(500) \\ Colin Barker, Jul 17 2014
(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)]
print(auptont(565)) # Michael S. Branicky, Nov 30 2021

Crossrefs

Cf. A007510.

Sequence in context: A271348 A057739 A146751 * A163060 A247433 A051682

Adjacent sequences: A162731 A162732 A162733 * A162735 A162736 A162737

Keyword

nonn,less

Author

Juri-Stepan Gerasimov, Jul 13 2009

Extensions

Replaced 55 by 60 and 447 by 446 - R. J. Mathar, Sep 23 2009

Status

approved