cp's OEIS Frontend

A071681 Number of ways to represent the n-th prime as arithmetic mean of two other primes.

%I A071681 #37 May 27 2023 10:55:59
%S A071681 0,0,1,1,2,2,3,1,3,3,2,4,4,4,4,5,5,3,5,7,5,4,5,6,6,8,6,7,6,6,8,8,10,6,
%T A071681 10,8,8,6,10,8,9,7,9,11,10,6,10,11,11,8,12,10,10,14,13,14,13,9,10,13,
%U A071681 12,12,14,16,11,13,13,14,18,13,18,14,14,17,14,16,14,16,15,16,16,17,16,16
%N A071681 Number of ways to represent the n-th prime as arithmetic mean of two other primes.
%C A071681 Conjecture: a(n)>0 for n>2.
%C A071681 a(A137700(n))=n and a(m)<>n for m < A137700(n), A000040(A137700(n))=A126204(n). - _Reinhard Zumkeller_, Feb 07 2008
%C A071681 The conjecture follows from a slightly strengthened version of Goldbach's conjecture: that every even number > 6 is the sum of two distinct primes. - _T. D. Noe_, Jan 10 2011 [Corrected by _Barry Cherkas_ and _Robert Israel_, May 21 2015]
%C A071681 a(n) = A116619(n) + 1. - _Reinhard Zumkeller_, Mar 27 2015
%C A071681 Number of primes q < prime(n), such that 2*prime(n) - q is prime. - _Dmitry Kamenetsky_, May 27 2023
%H A071681 Reinhard Zumkeller, <a href="/A071681/b071681.txt">Table of n, a(n) for n = 1..10000</a>
%H A071681 Mladen Vassilev-Missana, <a href="http://www.nntdm.net/papers/nntdm-11/NNTDM-11-1-20-22.pdf">Goldbach's n-perfect numbers as a key for proving the Goldbach's Conjecture</a>, Notes on Number Theory and Discrete Mathematics (2005) Vol. 11, No. 1, 20-22.
%e A071681 a(7)=3 as prime(7) = 17 = (3+31)/2 = (5+29)/2 = (11+23)/2 and 2*17-p is not prime for the other primes p < 17: {2,7,13}.
%t A071681 f[n_] := Block[{c = 0, k = PrimePi@n - 1}, While[k > 0, If[ PrimeQ[2n - Prime@k], c++ ]; k-- ]; c]; Table[ f@ Prime@n, {n, 84}] (* _Robert G. Wilson v_, Mar 22 2007 *)
%o A071681 (PARI) A071681(n)={s=2*prime(n);a=0;for(i=1,n-1,a=a+isprime(s-prime(i)));a}
%o A071681 (Haskell)
%o A071681 a071681 n = sum $ map a010051' $
%o A071681    takeWhile (> 0) $ map (2 * a000040 n -) $ drop n a000040_list
%o A071681 -- _Reinhard Zumkeller_, Mar 27 2015
%Y A071681 Cf. A071680, A000040, A129363, A178609, A001358, A100484, A001747, A010051, A116619, A253138.
%K A071681 nonn,look
%O A071681 1,5
%A A071681 _Reinhard Zumkeller_, May 31 2002