A137700 -id:A137700 - cp's OEIS frontend

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

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, 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, 12, 12, 14, 16, 11, 13, 13, 14, 18, 13, 18, 14, 14, 17, 14, 16, 14, 16, 15, 16, 16, 17, 16, 16

Offset: 1

Reinhard Zumkeller, May 31 2002

Conjecture: a(n)>0 for n>2.
a(A137700(n))=n and a(m)<>n for m < A137700(n), A000040(A137700(n))=A126204(n). - Reinhard Zumkeller, Feb 07 2008
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]
a(n) = A116619(n) + 1. - Reinhard Zumkeller, Mar 27 2015
Number of primes q < prime(n), such that 2*prime(n) - q is prime. - Dmitry Kamenetsky, May 27 2023

			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}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mladen Vassilev-Missana, Goldbach's n-perfect numbers as a key for proving the Goldbach's Conjecture, Notes on Number Theory and Discrete Mathematics (2005) Vol. 11, No. 1, 20-22.

Cf. A071680, A000040, A129363, A178609, A001358, A100484, A001747, A010051, A116619, A253138.

a071681 n = sum $ map a010051' $
   takeWhile (> 0) $ map (2 * a000040 n -) $ drop n a000040_list
-- Reinhard Zumkeller, Mar 27 2015

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 *)

A071681(n)={s=2*prime(n);a=0;for(i=1,n-1,a=a+isprime(s-prime(i)));a}

A126204 Least prime representable as the arithmetic mean of two other primes in n different ways.

Original entry on oeis.org

2, 5, 11, 17, 37, 53, 89, 71, 101, 179, 137, 193, 233, 257, 251, 401, 311, 373, 347, 457, 661, 479, 547, 599, 617, 641, 683, 907, 881, 773, 827, 809, 941, 1033, 977, 1289, 1201, 1187, 1229, 1223, 1277, 1361, 1597, 1493, 1447, 1499, 1451, 1549, 1487, 1709, 1933

Offset: 0

Robert G. Wilson v, Mar 22 2007

nonn

a(n)=A000040(A137700(n)); A071681(A137700(n))=n and A071681(m)<>n for m < A137700(n). - Reinhard Zumkeller, Feb 07 2008

Robert G. Wilson v, Table of n, a(n) for n = 0..11300

Cf. A071681.

f[n_] := Block[{c = 0, k = PrimePi@n - 1}, While[k > 0, If[ PrimeQ[2n - Prime@k], c++ ]; k-- ]; c]; t = Table[0, {1000}]; Do[ p = Prime@n; a = f@p; If[a < 1001 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]], {n, 8900}]

Start of b-file modified to match sequence by N. J. A. Sloane, Aug 31 2009

cp's OEIS Frontend

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

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