A126204 -id:A126204 - 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}

A136244 Least positive integer k such that 2k can be expressed as the sum of two primes in exactly n ways.

Original entry on oeis.org

1, 2, 5, 11, 17, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651, 540, 615, 759, 525, 570, 693, 645

Offset: 0

K. B. Subramaniam (shunya_1950(AT)yahoo.co.in), Dec 24 2007

nonn

It appears that 2, 3, 4, 6 are the only numbers k such that 2k can be expressed as the sum of two primes in only one way.

Except when n = 1, a(n) = A258713(n). The first 11 terms of this sequence are the same as the initial terms of A053033. If a(n) exists for all n then A053033 is a subsequence. - Andrew Howroyd, Jan 28 2020

			a(3) = 11: 22 = 3 + 19 = 5 + 17 = 11 + 11. Also 22 is the least number which could be expressed as the sum of two prime numbers in exactly three ways.

David A. Corneth, Table of n, a(n) for n = 0..16805 (first 1001 terms from Andrew Howroyd)
Index entries for sequences related to Goldbach conjecture

Cf. A023036, A045917, A053033, A126204, A258713.

a(n, lim=oo)={for(i=1, lim, my(s=0); forprime(p=2, i, s+=isprime(2*i-p)); if(s==n, return(i))); -1} \\ Andrew Howroyd, Jan 28 2020

From Andrew Howroyd, Jan 28 2020: (Start)
a(n) = A023036(n) / 2.
A045917(a(n)) = n. (End)

a(0)=1 prepended, a(5) corrected and a(7) and beyond from Andrew Howroyd, Jan 28 2020

A137700 Smallest k such that the k-th prime has exactly n distinct representations as arithmetic mean of two primes.

Original entry on oeis.org

1, 3, 5, 7, 12, 16, 24, 20, 26, 41, 33, 44, 51, 55, 54, 79, 64, 74, 69, 88, 121, 92, 101, 109, 113, 116, 124, 155, 152, 137, 144, 140, 160, 174, 165, 209, 197, 195, 201, 200, 206, 218, 251, 238, 229, 239, 230, 244, 236, 267, 295, 281, 299, 301, 307, 312, 313, 325

Offset: 0

Reinhard Zumkeller, Feb 07 2008

nonn

A071681(a(n))=n and A071681(m)<>n for mA000040(a(n))=A126204(n).

Corrected description suggested by Zak Seidov, May 16 2008

A222590 Greatest prime representable as the arithmetic mean of two other primes in n different ways, or 0 if no such prime exists.

Original entry on oeis.org

3, 19, 31, 61, 79, 83, 199, 181, 229, 271, 277, 313, 293, 439, 389, 401, 499, 619, 601, 709, 859, 643, 787, 811, 743, 823, 1039, 1009, 1321, 1021, 1279, 1213, 1249, 1489, 1483, 1301, 1609, 1621, 1459, 1753, 1559, 1877, 2011, 2029, 1741, 1901, 2087, 2239, 2207

Offset: 0

Robert G. Wilson v, Feb 25 2013

nonn

a(6681) is probably the only such term which equals zero.

			There are only two primes which are not the arithmetic mean of two other primes and they are 2 and 3. Therefore a(0)=3.
There are only three primes which are the arithmetic mean of two other primes in just one way. They are 5 = (3+7)/2, 7 = (3+11)/2, and 19 = (7+31)/2. Therefore a(1)=19.
There are only three primes which are the arithmetic mean of two other primes in just two ways. They are 11 = (3+19)/2 = (5+17)/2, 13 = (3+23)/2 = (7+19)/2, and 31 = (3+59)/2 = (19+43)/2. Therefore a(2)=31, etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Goldbach conjecture

Cf. A071681, A126204.

f[n_] := Block[{c = 0, k = 2, p = Prime@ n}, While[k + 1 < p, If[PrimeQ[p - k] && PrimeQ[p + k], c++ ]; k += 2]; c]; t = Table[0, {1000}]; Do[a = f@ n; If[a < 1001, t[[a + 1]] = Prime@ n; Print[{a, Prime@ n}]], {n, 5000}]; Take[t, 50]

A157478 a(n) is the least prime p such that p is greater than any previous term and is representable as the arithmetic mean of two other primes in exactly n different ways.

Original entry on oeis.org

5, 11, 17, 37, 53, 89, 107, 127, 179, 197, 223, 233, 257, 263, 401, 409, 421, 449, 457, 661

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

A number p is representable as the arithmetic mean of two other primes in n ways if there are n values of k such that p + k and p - k are both prime.
If the restriction that a(n) must be greater than previous terms is removed then the sequence would be A126204. - Andrew Howroyd, Jan 12 2020
The next term if it exists > 10^6. - Andrew Howroyd, Jan 12 2020

			a(1) = 5 because 5+-2 are primes.
a(2) = 11 because 11+-6, 11+-8 are primes.
a(3) = 17 because 17+-6, 17+-12, 17+=14 are primes.
a(4) = 37 because 37+-6, 37+-24, 37+-30, 37+-34 are primes.
a(5) = 53 because 53+-6, 53+-30, 53+-36, 53+-48, 53+-50 are primes.

Cf. A126204.

q=1;lst={};Do[p=Prime[n];i=0;Do[If[PrimeQ[p-k]&&PrimeQ[p+k],i++;],{k,2,p,2}];If[i==q,AppendTo[lst,p];q++ ],{n,2*5!}];lst

a(n, lim=oo)={my(v=vector(n),r=1); forprime(p=5, lim, my(k=0); forprime(q=3, p-2, k+=isprime(2*p-q)); if(k==r, if(r==n, return(p)); r++))} \\ Andrew Howroyd, Jan 12 2020

Definition clarified by Andrew Howroyd, Jan 12 2020

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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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A136244 Least positive integer k such that 2k can be expressed as the sum of two primes in exactly n ways.

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A137700 Smallest k such that the k-th prime has exactly n distinct representations as arithmetic mean of two primes.

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A222590 Greatest prime representable as the arithmetic mean of two other primes in n different ways, or 0 if no such prime exists.

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A157478 a(n) is the least prime p such that p is greater than any previous term and is representable as the arithmetic mean of two other primes in exactly n different ways.

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