A071680 -id:A071680 - 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}

A085704 Primes of the form prime(k)*2 - prime(k-1), k>1.

Original entry on oeis.org

7, 43, 59, 73, 163, 179, 223, 263, 269, 283, 379, 491, 569, 599, 613, 619, 643, 659, 739, 821, 953, 983, 1021, 1109, 1129, 1193, 1229, 1303, 1373, 1483, 1523, 1613, 1669, 1733, 1753, 1759, 1789, 1873, 1913, 1913, 1949, 1999, 2143, 2293, 2383

Offset: 1

Reinhard Zumkeller, Jul 18 2003

nonn

Zak Seidov, Table of n, a(n) for n = 1..2000.

Cf. A071680, A000040.

A341284 a(n) is the least prime == -prime(n) (mod 2*prime(n+1)).

Original entry on oeis.org

7, 23, 37, 41, 89, 59, 73, 151, 157, 43, 127, 131, 239, 59, 419, 307, 73, 359, 367, 401, 419, 1163, 881, 307, 311, 967, 547, 569, 3697, 397, 691, 419, 457, 757, 163, 821, 839, 179, 1259, 907, 2111, 967, 1777, 599, 223, 3803, 3863, 2063, 3499, 1201, 3617, 2269, 263, 269, 1889, 2441, 283, 1409

Offset: 2

J. M. Bergot and Robert Israel, Feb 25 2021

a(k) is the least odd prime == -prime(k) (mod prime(k+1)).
a(k) = A163981(k) if and only if k is not in A029707.
a(k) = 2*prime(k+1)-prime(k) if and only if prime(k+1) is in A071680.

			a(3) = 23 is the least prime == -5 (mod 14), where prime(3) = 5 and prime(4) = 7.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A029707, A071680, A163981, A342027.

f:= proc(n) local k;
  for k from 2*ithprime(n+1)-ithprime(n) by 2*ithprime(n+1)  do
    if isprime(k) then return k fi
  od;
end proc:
map(f, [$2..100]);

a(n) = forprime(p=2,, if (Mod(p, 2*prime(n+1)) == -prime(n), return (p))); \\ Michel Marcus, Feb 25 2021

(a(k) + prime(k)) mod (2*prime(k+1)) = 0.

A244374 Primes which are the arithmetic mean of two consecutive primes of the form 4n+3.

Original entry on oeis.org

5, 37, 53, 157, 173, 233, 257, 277, 353, 373, 401, 593, 613, 653, 733, 769, 977, 1097, 1297, 1433, 1493, 1613, 1753, 1993, 2137, 2161, 2377, 2417, 2677, 2693, 2749, 2797, 3313, 3533, 3637, 3733, 4013, 4133, 4457, 4513

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2014

nonn

All terms must necessarily be primes of form 4n+1. - Jens Kruse Andersen, Jul 15 2014

			5 is in this sequence because (A002145(1) + A002145(2))/2 = (3 + 7)/2 = 5 and 5 is prime.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A071680.

N:= 10000; # get all entries <= N
P3:= select(isprime,[seq(4*i+3,i=0..floor((N-3)/4))]):
select(isprime,[seq((P3[i]+P3[i+1])/2, i=1..nops(P3)-1)]); # Robert Israel, Jul 15 2014

p=[]; forstep(n=3, 5000, 4, if(isprime(n), p=concat(p, n))); p;
s=[]; for(k=1, #p-1, if(isprime(q=(p[k]+p[k+1])\2), s=concat(s, q))); s \\ Colin Barker, Jun 27 2014

One term deleted and another term inserted by Colin Barker, Jun 27 2014

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

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A085704 Primes of the form prime(k)*2 - prime(k-1), k>1.

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A341284 a(n) is the least prime == -prime(n) (mod 2*prime(n+1)).

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A244374 Primes which are the arithmetic mean of two consecutive primes of the form 4n+3.

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