A071681 -id:A071681 - cp's OEIS frontend

A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

3, 3, 5, 7, 11, 7, 17, 17, 19, 31, 29, 19, 41, 47, 47, 43, 61, 59, 67, 61, 59, 71, 67, 89, 97, 101, 79, 89, 103, 113, 107, 127, 131, 139, 151, 127, 137, 167, 167, 163, 149, 163, 167, 157, 199, 163, 197, 181, 227, 227, 211, 239, 251, 257, 257, 229, 271, 269

Offset: 1

Giovanni Teofilatto, May 15 2007

The same selection rule as in A078497 applies: if there is more than one prime triple (p,q=p+d,r=q+d) with p+q+r=A001748(n), take p from the triple with minimum d. - R. J. Mathar, May 19 2007

			3 + 5 + 7 = 15, which is three times the (1+2)th prime number. Thus a(1) = 3.

Cf. A078497, A071681, A078611.

A129758 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(p) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A129758(n)) ; od ; # R. J. Mathar, May 19 2007

a[n_]:=Module[{},k=1; While[Not[PrimeQ[Prime[n+1]-k] && PrimeQ[Prime[n+1]+k]], k++ ]; Prime[n + 1] - k]; Table[a[n], {n, 2, 60}]

A078497(n)-prime(n)=prime(n)-a(n)=d. - R. J. Mathar, May 19 2007

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n+2)) = 1. - Alain Rocchelli, May 01 2024

Edited and extended by R. J. Mathar, Giovanni Teofilatto and Stefan Steinerberger, May 19 2007

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]

A268914 Minimum difference between two distinct primes whose sum is 2*prime(n), n>4.

Original entry on oeis.org

12, 12, 12, 24, 12, 24, 24, 12, 24, 48, 12, 12, 24, 36, 12, 24, 12, 36, 48, 36, 60, 24, 12, 12, 60, 48, 48, 36, 60, 24, 36, 24, 12, 72, 60, 12, 24, 36, 84, 60, 60, 84, 24, 120, 60, 96, 12, 24, 60, 24, 12, 12, 24, 84, 12, 24, 108, 48, 48, 84, 72, 72, 36, 60, 72, 36, 12, 84, 60, 12, 60, 72, 60, 48, 36, 24, 60, 24, 24, 48, 36, 48, 36, 168, 36, 48

Offset: 5

Barry Cherkas, Feb 15 2016

If p>4 is prime, any two primes that add to 2p must be equidistant from p. If p is congruent to 1 Mod 3, then p+2 and p-4 are divisible by 3. Alternatively, if p is congruent to 2 Mod 3, the p-2 and p+4 are divisible by 3. Thus, the equidistant pairs (p-2,p+2) and (p-4,p+4) cannot be primes that add to 2p. On the other hand, adding or subtracting any multiple of 6 will be congruent to the same congruence class as p and may be prime. Thus, the minimal difference between distinct primes that add to p must be a multiple of 12.

Extrapolating from computational evidence for all primes up to 10^9, we conjecture: For each multiple of 12 there are infinitely many primes p such that p-6k and p+6k are prime and 12k is the minimal difference for two distinct primes whose sum is 2p.

			For n=5, 2*prime(5)=2*11=5+17 and 17-5=12.
For n=6, 2*prime(6)=2*13=7+19 and 19-7=12.
...
For n=8, 2*prime(8)=2*19=7+31 and 31-7=24.

Barry Cherkas, Table of n, a(n) for n = 5..10003
G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, 1923.

Cf. A078611, A071681, A139602, A006489, A137796, A161723, A128940.

N:= 1000: # to get a(5) .. a(N)
p:= 7:
for n from 5 to N do
  p:= nextprime(p);
  for k from 6 by 6 while not isprime(p+k) or not isprime(p-k) do od:
  A[n]:= 2*k
od:
seq(A[n],n=5..N); # Robert Israel, Mar 09 2016

f[n_]:=Block[{p=Prime[n],k},k=p+6;
While[!PrimeQ[k]||!PrimeQ[2p-k],k=k+6];2(k-p)];
seq=Reap[Do[Sow[f[n]],{n,5,200}]][[2]][[1]];
seq
(*For large data sets (say, N>5000), replace 200 with N and the above algorithm is comparatively efficient.*)
Table[2 SelectFirst[Range[#/2], Function[k, AllTrue[{#/2 + k, #/2 - k}, PrimeQ]]] &[2 Prime@ n], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

a(n) = {p = prime(n); d = 2; while (! (isprime(p-d) && isprime(p+d)),  d+=2); 2*d;} \\ Michel Marcus, Mar 17 2016

a(n) = 2*A078611(n+2).

A359556 Number of ways to represent the average of the n-th twin prime pair as arithmetic mean of the averages of two other twin prime pairs.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 6, 1, 2, 0, 1, 3, 4, 2, 3, 0, 7, 2, 3, 1, 4, 4, 1, 3, 6, 5, 1, 1, 3, 4, 6, 1, 11, 6, 7, 3, 6, 2, 10, 5, 4, 4, 6, 4, 2, 1, 7, 1, 4, 5, 4, 4, 4, 8, 7, 2, 3, 4, 3, 3, 10, 9, 3, 5, 21, 17, 5, 12, 5, 2, 3, 3, 18, 13, 4, 19, 11, 15, 5

Offset: 1

Tamas Sandor Nagy, Jan 05 2023

nonn

			a(4) = 1 because 18, the average of the 4th twin prime pair (17, 19), can be expressed in one way only as the arithmetic mean of the averages of two other twin prime pairs. These are (5, 7) and (29, 31) with their averages 6 and 30: (6 + 30)/2 = 36/2 = 18.
a(7) = 2 because 60, the average of the 7th twin prime pair (59, 61), can be expressed in two ways as the arithmetic mean of the averages of two other twin prime pairs. Firstly, by the averages 12 and 108 of the twin prime pairs (11, 13) and (107, 109), since (12 + 108)/2 = 120/2 = 60. Secondly, by the averages 18 and 102 of the twin prime pairs (17, 19) and (101, 103), as (18 + 102)/2 = 120/2 = 60 also.
a(15) = 0 because 198, the average of the 15th twin prime pair (197, 199), cannot be expressed as the arithmetic mean of the averages of any other two twin prime pairs.

Amiram Eldar, Table of n, a(n) for n = 1..1200

Cf. A014574, A071681.

means = Select[2*Range[3500], PrimeQ[# - 1] && PrimeQ[# + 1] &]; Count[(Plus @@@ Subsets[means, {2}])/2, #] & /@ Select[means, # < Max[means]/2 &] (* Amiram Eldar, Jan 06 2023 *)

More terms from Amiram Eldar, Jan 06 2023

A086712 Number of times the n-th prime power can be written as an arithmetic mean of two other prime powers.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 4, 3, 4, 6, 6, 5, 5, 7, 8, 5, 6, 6, 7, 6, 8, 6, 8, 8, 7, 8, 10, 10, 9, 8, 9, 14, 8, 10, 11, 10, 12, 8, 11, 8, 12, 13, 12, 11, 11, 13, 13, 13, 13, 13, 11, 11, 14, 11, 13, 16, 12, 16, 14, 15, 16, 17, 13, 16, 15, 12, 18, 27, 15, 19, 18, 17, 15, 16, 15, 13, 18, 17, 15

Offset: 1

Reinhard Zumkeller, Aug 01 2003

nonn

			n=7, A000961(7)=8=2^3: (3+13)/2=(A000961(3)+A000961(10))/2, (5+11)/2=(A000961(5)+A000961(9))/2 and (7+3^2)/2=(A000961(6)+A000961(8))/2: therefore a(7)=3.

Eric Weisstein's World of Mathematics, Prime Power

Cf. A000961, A071681.

A137670 Prime numbers p such that p-b < p-a < p < p+a < p+b are prime for some a and b.

Original entry on oeis.org

17, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 27 2008; corrected May 02 2008

nonn

It seems highly likely that all primes other than 2,3,5,7,19 are in this sequence. There are no further exceptions to 4 billion. - Charles R Greathouse IV, Apr 19 2010

Cf. A071681.

s=""; q=1; For[i=2, i<10^2, p=Prime[i]; For[a=2, aq&&PrimeQ[p-c]&&PrimeQ[p+c], (*Print[p, ":", a, ", ", b, ", ", c]; *)s=s<>ToString[p]<>", "; q=p]; c=c+2]]; b=b+2]]; a=a+2]; i++ ]; Print[s]

New name from Charles R Greathouse IV, Apr 19 2010

cp's OEIS Frontend

A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

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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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A268914 Minimum difference between two distinct primes whose sum is 2*prime(n), n>4.

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A359556 Number of ways to represent the average of the n-th twin prime pair as arithmetic mean of the averages of two other twin prime pairs.

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A086712 Number of times the n-th prime power can be written as an arithmetic mean of two other prime powers.

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A137670 Prime numbers p such that p-b < p-a < p < p+a < p+b are prime for some a and b.

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